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Initials/Date: _LDS /_3-8-2007 
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/02^-N/<*5~-fH< NUREG/CR-3332 

f

 ' ORNL-5968 

! 

Radiological Assessment 

A Textbook on Environmental Dose Analysis -

Date Published: September 1983 

Edited by 

John E. Till 

and 

H. Robert Meyer 

"TITtfety ReSearch Di^on 

Oa* Hi Nr?ional Laboratory 

ualcRidge, Tennessee . -- "" ' 

Prepared for 

fcfrt Integration 

U

-S• NucWp31" «

e

?

c t w

 Regulation 

n

*l* FfN B0766 
4-1 

4 Transport of Radionuclides 

in Groundwater 

ByR. B. CODELL* and J. D. DUGUID* 

1 

4,1 INTRODUCTION 

Groundwater flow is one of the likely pathways for radionuclides released 

from waste disposal areas. Groundwater transport is also a major pathway for 

certain classes of accidental and normal releases from nuclear facilities such as 

power plants, fuel reprocessing plants, arid mining or. milling operations. The 

primary emphasis of this chapter will be "groundwater transport of 

radionuclides using waste disposal as an application. Concentrations of 

radionuclides that could reach the biosphere and the resulting consequences 

must be predicted using scenarios.of events and processes that are possible but 

unlikely to occur at the disposal" site. Estimations of groundwater flow and 

transport are important in assessing the performance of a disposal system 

because they are probable migration pathways between the nuclear waste and 

the biosphere. 

The transport of radionuclides through the ground can be estimated by 

•the use of tracers, groundwater dating, mathematical models, or. by a 

combination, of all of the above. Chemical or .radioactive tracers may be 

deliberately introduced to the groundwater and monitored through wells for the 

direct determination of. groundwater velocity and: transport. Alternatively, 

pollutants not deliberately introduced to the groundwater may also be traced. 

Groundwater dating is a technique by which the age of a region of 

groundwater may be estimated from the concentration of an atmospheric 

*U.S. Nuclear Regulatory Commission, Hydrologic Engineering Section, Washington, D.C. 

Battelle Office of National Waste Terminal Storage Integration, Germantown, 

Md. 
4-2 Radiological Assessment 

radionuclide it contains. Tritium released since the beginning of the nuclear 

era can be used to date water up to several decades. Water may be dated using 

l4C over longer periods of hundreds to thousands of years. Direct 

measurements of the migration of radionuclides released from naturally 

occurring uranium and thorium ore bodies can be used as a close analog to 

man-made radioactive waste disposal situations. 

Using groundwater flow and transport models provides a means to 

calculate the expected concentrations of radionuclides following release to the 

environment. Where groundwater contamination of wells or surface water 

bodies such as lakes, streams, or rivers occurs, the radionuclide concentrations 

are used in other pathway models to calculate the consequences of the release. 

Pathway models consist of surface water transport models and biological 

pathway models, which in turn provide the basis for dose calculations. Doses to 

humans arise through the contamination of drinking water and food and from 

contaminated surfaces such as flood plains and beaches. 

This chapter discusses current practice in groundwater flow and transport 

modeling as well as data requirements and possible misuses of models. A set of 

analytical models is presented along with illustrations of their use. 

4.1.1 Types of Groundwater Assessments . ;, . ' 

4.1.1.1 Geologic Isolation of High-Level Waste 

Actual tests and demonstrations of the behavior pf a high-level waste 

(HLW) repository system cannot be perfprmed oyer t}ie operational lifetime of 

the repository. Therefore, we must rely on mathematical models for 

performance assessments, using data collected over comparatively short periods 

of time, to predict the long-term performance of the system.' This is the only 

means by which the cumulative effects of changes in the properties of the 

respository system, the effects of various design features of the repository, and 

the effects of the repository on the environment can be analyzed. Performance 

assessment not only provides this type of analysis but also provides information 

that is useful in guiding research and development activities in site selection, 

repository design, and waste package design. Performance assessment treats 

concepts that can be quantified,r that is, failure analysis and consequence 

assessment (Klingsberg and Duguid 1980). 

An assessment of the long-term performance of a repository analyzes the 

events and processes that could release radionuclides from the waste and the 

phenomena that might transport • radionuclides to the biosphere. These 

phenomena may be roughly classified as those that occur in the near field 

(where waste and repository phenomena dominate) and those that occur in the 

far field (at a greater distance from the repository where natural phenomena 

dominate). Although these two regions are not separated by a precisely defined 

Radionuclides in Groundwater 4-3 

boundary, the distinction is useful because the physical and chemical effects of 

heat and radiation from the waste are limited to the near field (Klingsberg and 

Duguid 1980). Different methods pf analysis are therefore appropriate for the 

two regions. Near-field analysis studies the combined effects of heat, radiation, 

repository design and construction, and the waste package. Far-field analysis 

studies the effects of events that arise from natural phenomena, and from 

potential human actions after the repository has been sealed. These far-field 

phenomena usually appear in the geosphere and the biosphere outside of the 

repository. Both near-field and far-field performance must be considered in 

determining how well the natural and the man-made components of the 

disposal system meet the criteria for repository performance. 

4.1.1.2 Shallow Land Burial 

For near-surface disposal such as shallow land burial, the analysis of 

system performance is similar to that of isolation of high-level waste with two 

major exceptions: (1) groundwater flow and transport models must frequently 

consider the unsaturated zone and (2) the wastes are not heat producing. The 

analysis proceeds in much the same fashion as for high-level waste and 

includes the development of a source term through corrosion or breaching of 

waste containers, defining an , appropriate leach rate for the waste form, 

developing a system release scenario, and calculating groundwater flow and 

radionuclide transport for use in the biosphere models and dose codes (Aikens 

etal. 1979). 

4.1.1.3 Uranium Mining and Milling 

There are several potential groundwater contamination problems 

associated with the mining, milling, and waste disposal operations necessary to 

produce uranium fuels (USNRC 1979; Shepard and Cherry 1980). 

, The greatest waterborne contamination hazard to groundwater is the 

seepage from tailings ponds resulting from conventional milling procedures. 

The waste stream contains about half solids and half water, which is usually 

disposed of in ponds (tailings ponds) formed behind earth or rubble dams. 

Tailings are sometimes reburied in the ore pits. Acid leach mills are the most 

prevalent type. Tailings ponds usually receive highly acidic (pH 0.5-2) water 

and tailings, but in some cases tailings are first neutralized. 

The wastes from the tailings ponds differ most from other forms of 

nuclear waste because of their unique chemistry. In acidic tailings, most of the 

radioactive and other chemical wastes are in the dissolved state. Acidic wastes 

are sometimes neutralized to reduce the solubility of pollutants, but in some 

cases the wastes slowly become apidic because of oxidation of pyrite (iron 

sulfide). The behavior of the radioactive contaminants varies from very simple 

to very complex. Probably the most radiologically significant radioactive waste 
4-4 Radiological Assessment 

component present is radium, which has a fairly simple Chemistry since it 

exists only in +2 valence state. Uranium and several other radioactive 

waste compounds behave in a much more complicated fashion, since they may 

exist in several different valence states and form complex compounds (Landa 

1980). . ' , 

The solubility of all of the contaminants is high; for low-pH conditions and 

decreases markedly at higher pH. Neutralization by carbonates, such as 

limestone, either added deliberately pr encountered in the environment, 

however, can mobilize uranium in the form of soluble, carbonate complexes. 

Uranium may also be mobilized in an oxidizing environment or by certain 

organic chemicals in groundwater (Shephard and Cherry 1980). 

4.1.1.4 Nuclear Power Plant Accidents 

Postulated accidental releases pf radioactivity to the groundwater pathway 

have been evaluated for a wide range of nuclear facilities either for generic 

sites or in actual reactor licensing reviews. The accidental releases considered 

range from small leaks from contaminated water streams in nuclear plants to 

major releases caused by a core meltdown accident (USNRC 1975; USNRC 

1978; Niemczyk 1981). 

Consideration given to. nuclear power plant accident releases to 

groundwater differs from those for high- and low-level waste disposal or other 

fuel cycle problems in several important respects: (1) The risk of 

contamination would exist only for the lifetime of the plant. Administrative 

controls would be in effect during this period, so mitigative measures could 

presumably be taken should an accidental release occur; (2) The isotopes of 

importance in nuclear power plant liquid pathway accidents are generally those 

with high dose factors and/or half-lives of years to tens of years, notably

 3H, 

134Cs, ,37Cs, 89Sr, ^Sr, and I06Ru. Unlike nuclear waste, long-lived 

radionuclides, actinides, and transuranics have been shown to be of much lower 

importance (USNRC 1978); (3) For a given event, consequences of radioactive 

release to the groundwater pathway1 typically present much smaller risks than 

release to the airborne pathway. These consequences should not be neglected in , 

citing studies for nuclear power plants, however. 

4.2 TYPES OF GROUNDWATER MODELS 

4.2.1 Groundwater Models for Low-Level Waste 

The assessment of a low-level waste burial, ground requires three types of 

models: (1) models to determine the portion of the radioactive source released, 

if infiltrating water contacts the waste, (2) mathematical models in terms of 

measurable hydrologic parameters that predict the migration.of radionuclides 

from the source to locations accessible to the public, and (3) models for 

Radionuclides in Groundwater 4-5 

determining the potential radiation dose using the radionuclide concentrations 

that reach accessible locations. In this chapter only item 2 will be discussed. 

Dose assessments are covered in other sections of this document. Source term 

definitions defy simple explanation and cannot be adequately covered in this 

report. 

Although difficult to analyze, the near field is as important and complex 

for shallow land disposal as for deep geological disposal. Most of the action is 

in the disposal trench and determines what is available for future groundwater 

transport. Determining the water balance and the amount of water infiltrating 

is difficult. Determining the leaching and release from chemically and 

physically heterogeneous wastes such as low-level wastes is even more difficult. 

To date, no model adequately addresses the problem of modeling the near-field 

environment for shallow land burial. 

The calculation of transport of radionuclides from shallow land burial sites 

is complicated by the waste frequently being leached in the unsaturated zone, 

and the movement of waste to the water table must consider both flow and 

transport through this zone. Here, in the simplest case, flow and transport may 

be assumed to be downward in one dimension. Water flow rates from a water 

balance can be used to approximate the unsaturated flow used as input to the 

radionuclide transport model. Radionuclide transport in the unsaturated zone 

can also be assumed to be downward and one dimensional. These simplifying 

assumptions can be used to calculate the concentrations of radionuclides being 

released to the water table as a function of time. More rigorous calculation? 

can also be done using two- and three-dimensional models of unsaturated flov, 

and transport. However, considering the uncertainty in the magnitude of thL^fl 

source term, these sophisticated calculations are warranted only when a large ^ 

amount of data is available and/or detailed results are required. Below the 

water table either numerical or analytical models of two- or three-dimensional 

groundwater flow and radionuclide transport can be used to calculate the 

concentration of radionuclides released at locations accessible to the public. 

4.2.2 Groundwater Models for High-Level Waste Repositories 

4.2.2.1 Far-Field Perfonnance 

Although there are significant differences in the state of development and 

verification of different far-field models, these models are sufficiently well 

advanced to be used in assessments of repository performance at either generic 

oj specific sites. In general, the procedure for calculating the far-field effects 

of a repository breach is shown in Fig. 4.1. 

After the scenarios that have to be modeled have been identified, the next 

step in a performance assessment is to predict their consequences. Whether the 

scenarios will actually occur cannot be predicted with complete certainty, but 

ro 

Ol 
4-6 Radiological Assessment 

ORNL-OWG 82-14405 

RELEASE. 

SCENARIO 

I 

SOURCE 

TERM 

I 

TRANSPORT 

• THROUGH 

GEOSPHERE 

I 

TRANSPORT 

.THROUGH,, 

BIOSPHERE 

I 

DOSE 

CALpULATIONS 

Figure 4.1. Elements of far-field risk assessment. Source: Klihgsberg, C, and Duguid, J. 

1980. Status of Technology for Isolating ffigti-Level Radioactive Waste in Geologic 

Repositories, U.S. Department of Energy, Tech. Inf. Cent., Oak Ridge, Tenn. ' 

where possible the probabilities of their occurrences , are estimated. 

Probabilities are, however, highly uncertain for the, events that have occurred 

in the region around a repository site only a few times in geologic history or 

that may not have occurred there at all. Fpr this reason, the assessment of 

repository performance relies heavily on predictions of the consequences of 

scenarios rather than on predictions of their probabilities. 

The source term describes the waste at all times; it specifies the 

radionuclides present and the physical and chemical conditions of the waste. 

The radionuclide concentrations at the time of the breach can be calculated 

from their original concentrations in the waste. 

Source term evaluation, is highly site-specific, depending on such factors as 

the chemistry of the waste, the host rock, and the groundwater. Also, 

interaction between natural and man-made components can play an important 

Radionuclides in Groundwater 4-7 

role in defining release mechanisms. Further discussion of this important 

aspect of waste-migration modeling is beyond the scope of this book 

(Klingsberg and Duguid 1980). 

Once the source term has been defined, transport through the geosphere is 

determined by modeling fluid flow into and away from a repository. The 

output from the geosphere transport codes is a prediction of the radionuclide 

concentrations reaching the biosphere as a function of time. As contaminants 

move with groundwater, they may be sorbed and thus retarded by the rocks 

through which they pass. The parts of the contaminant transport models that 

describe sorption generally assume equilibrium between the concentration of 

contaminants in the fluid and the concentration on the rock surfaces. 

Because models of flow through fractures tend to be specific to particular 

types of fracture systems, they, are less universally applicable than porous 

media models (Duguid and Lee 1977). There are two basic problems for the 

modeling of material transport in fractured media. One problem is to assemble 

, .sufficient data to be able'to adequately describe the hydrology of the far-field 

region surrounding a repository site. The determination of effective 

permeabilities and fracture connections, is difficult, and a considerable amount 

of research remains before reliable methods will be available. The second 

problem is to understand the sorption process in the fractured rocks; although 

sorption is effective in porous rocks, it might be much less effective in 

fractured rocks. Successful modeling of flow in fractured aquifers is extremely 

limited. Modeling of radionuclide transport in fractured aquifers is nonexistent 

at the present time. Current models for porous media are being used with 

equivalent. formulations and conservative assumptions to establish bounds on 

. the effects of flow through fractured media (Klingsberg and Duguid 1980). 

4.2.2.2 Near-Field Performance 

Models for assessing the performance of a high-level waste repository in 

the near field must take into account mechanical stresses, heat flow, chemical 

1

 interactions, and radiation-induced physical-chemical processes. All of these 

phenomena, in addition to the properties of the host rock, affect the 

enyironment of the emplaced waste. 

The following sections discuss three principal types of models required for 

near-field analysis: heat transfer models, thermomechanical models, and 

models of physical and chemical interactions among the emplaced waste, the 

components of the waste package, and the host rock. 

Heat Transfer Models. Thermal models based on physical laws provide an 

accurate portrayal of heat flow and changes in temperature. The experimental 

results to date suggest that predictions of temperature within a few percent of 

measured values can be achieved. Consideration of heat transfer is important 

, because temperature gradients can be a driving force in groundwater flow. 
4-8 Radiological Assessment 

Over 40 such models are identified as useful in the studies of waste disposal 

(SAI 1979). 

Thermomechanical Models. The models are based on relations derived 

from laws of physics and functional relationships between stress and strain that 

are obtained from laboratory tests. Because repository rocks are 

inhomogeneous and may be fractured, generic functional laws are more ' 

difficult to obtain for rocks than for most other construction materials. 

Thermomechanical codes are currently being used, to analyze the uplift 

and subsidence, room stability and rate of closure, hole stability and rate of 

closure, canister movement, pillar stability, thermomechanical effects on' 

groundwater flow, stresses and strains at critical locations in the rock mass, 

and mechanical failure of the rock mass. These, phenomena arc rriost important 

to flow in fractured rocks. Current research emphasis is on the relationship 

between permeability and change'in functional geometry due to stress (SiAI 

Chemical Models. To predict the near-field behavior of a repository 

requires analyses of the interactions between the, tertiplaced package1 

components and the host rock.' These interactions fall into six general 

categories: (1) the movement of fluids in the vicinity of the waste package, (2) 

the corrosion of the canister and sleeve materials by these fluids, (3) the 

dissolution of the waste form by groundwater containing the added corrosion 

products, (4) the sorption of radionuclides by the rocks and the engineered 

components of the repository, (5) the absorption of radiation emitted by the 

waste, and (6) the alteration of chemical phases and properties in the vicinity , 

of the canisters. A study of these interactions predicts the kinds, amounts, and 

chemical state of the radionuclides available for entry into the groundwater 

system (Jenne 1979). 

4.2.3 Groundwater Models for Mill Tailings Waste Migration 

'i 

The modeling of transport from mill tailings is similar in many ways tp 

other types of groundwater waste migration problems such as low-level 

radioactive waste disposal. The unique aspects of the modeling of mill tailings 

wastes are the complex chemistry of the wastes and the process of 

neutralization, especially by rocks in the transport pathway. Concentrations of 

chemicals may be quite high in a mill tailings pond, which complicates the 

transport processes. Typical equilibrium concepts such as the retardation factor 

and Kj will not work well for complicated, nonlinear phenomena such as 

precipitation, which may be particularly severe at high concentrations. 

Unsaturated flow in some of the pond settings and the transient existence of 

the milling operations may present special modeling problems (Shepard and 

Cherry 1980; USNRC 1979). 

Radionuclides in Groundwater 4-9 

. 4.3 EQUATIONS FOR GROUNDWATER FLOW 

AND RADIOACTIVITY TRANSPORT 

The movement of radionuclides in groundwater can be described by two 

equations: one for the.niovement of the carrier fluid (water) and one for the 

mass transport of the dissolved constituents (radionuclides). In using these 

, equations, the movement of the carrier in the region under consideration must 

be known before the transport equation can be solved. 

The following discussion of the equations associated with the movement of 

groundwater and the transport of dissolved radioactive substances can be used 

only as a general guide. A person intending to use models to analyze a specific 

problem would usually need the.aid of an experienced groundwater hydrologist 

and modeler. A set of highly simplified transport models is presented in Sect. 

4.5.3, along with examples of their use. 

4.3.1 Groundwater Flow 

Radioactive releases may travel in the unsaturated region (e.g., region 

aboveithe water table) before entering the zone of saturation (e.g. below water 

table). However,.the release can also be directly into the zone of saturation. 

The predominant direction of the unsaturated flow is downward until the flow 

1

 reaches the zone of saturation. Within the zone of saturation the flow is 

predominantly lateral. 

The governing equations in the unsaturated zone consist of a set of 

coupled equations for the movement of gas and water. To date, only computer 

codes of limited applicability are available for the solution of these coupled 

gas-water equations (Lappala 1981). When the assumptions are made that the 

water moves as a single phase and that no trapped air pockets exist, a single 

governing equation for saturated-unsaturated flow is obtained (ANS 1980). 

n' dn ^ = V-(*(/«)(VA+Vz)), (4.1) 

where 

n — 

i 

a — 

0-

h = 

t = 

k = 

z = 

V = 

the moisture content (dimensionless), 

total porosity (dimensionless), 

the modified coefficient of compressibility of the medium (1/cm), 

the modified coefficient of compressibility of water (1/cm), 

the pressure head (cm), 

the time (s), 

the hydraulic conductivity tensor (cm/s), 

the elevation head (cm), 

the del operator. 
4-10 Radiological Assessment Radionuclides in Groundwater 4-11 

Equation 4.1 is nonlinear because, for unsaturated flow, both hydraulic 

conductivity and moisture content are functions of pressure head. 

The solution of Eq. 4.1 in three dimensjpns is generally impractical: 

Simplifications must usually be fpund. Depending on the nature of the 

problem, analytical or numerical methods like the ones described in Reeves and 

Duguid (1975) and Reeves et al. (1977) can be used to analyze saturatedunsaturated flow. 

The hydraulic conductivity is a tensor that accounts for directional 

properties (anisotropy) that arise in formations such as layered sediments (i.e., 

hydraulic conductivity is different in different directions). If the coordinate 

system is oriented parallel to the principal components of hydraulic 

conductivity, only the principal components of the tensor are required. If the 

medium is further assumed to be homogeneous and isotropic (i.e., properties pf 

the medium are not direction dependent), hydraulic conductivity becomes a 

scalar and Eq. 4.1 becomes (ANS 1980) 

Ss dH ' ' (4.2) 

K dt ' 

where 

V

2H = 

H = the total head = h + z (cm), 

S, = pg (a + n'fl) = Specific Storage Coefficient (1/cm), 

p = the water density (g/cm3), 

g = the acceleration of gravity (cm/s2), , 

a = the coefficient of compressibility of the medium (cm-s2/g), 

0 = the coefficient of compressibility of water (cm-s2/g). 

This equation is valid for saturated flow in confined aquifers. For a 

confinea aquifer of thickness b, the storage coefficient and transmissivity are 

respectively defined as 

and Eq. 4.2 becomes 

S = S,b and T = Kb 

V-H = A M 

T dt 

(4.3) 

(4.4) 

In simulations using Eq. .4.4, the boundary conditions of leakage should be 

used when appropriate. For problems involving leaky aquifers, methods like 

those described by Bredehoeft and Pinder (1970) can be, used. 

For unconfined aquifers where compressibility of the medium and the 

water is relatively unimportant compared to the vertical' movement of the free 

surface (water table), the continuity equation can be written as follows (ANS 

1980): 

V

2

tf2 = 

2Sy dH 

K dt 

(4.5) 

where Sy is the specific yield of the aquifer (dimensionless). 

For specific yield of steady flow in either confined or unconfined aquifers, 

Eqs. 4.4 and 4.5 reduce respectively to the following: 

V

2H = 0 

V

2

#

2

 = 0. 

(4.6) 

(4.7) 

For simplified cases, analytical solutions of Eqs. 4.4, 4.5, 4.6, and 4.7 such as 

those given in Carslaw and Jaeger (1959) can be used. For more complex 

situations, numerical solutions such as those described in Gray et al. (1977) 

should be used; 

An approximation of the flux (volume of flow per unit cross-sectional 

area) in the major flow direction can be obtained using Darcy's law: 

Vx = -K^P- ~ -K 

dx 

AH 

Ax 

(4.8) 

where AH/Ax is the hydraulic gradient in the direction of flow. 

This approximation is crude but in many cases it is acceptable because of 

the inability to accurately measure spatial variations in the hydraulic 

conductivity. Use of this equation assumes a homogeneous isotropic medium in 

which the gradient is constant over the increment. The actual velocity of a 

nonadsorbed tracer would be greater than the flux since water is moving only 

in the pore spaces. The pore velocity (or seepage velocity), U, may be approximated by dividing the flux by the effective porosity: 

U = Vx/ne. (4.8a) 

• Example 4.1. For saturated groundwater flow, calculate the pore or 

seepage velocity in an "average" sandstone under a gradient of 0.01 cm/cm. 

Use arithmetic mean values in tables. 

Equation 4.8a applies. The arithmetic mean hydraulic conductivity, K, is 

3.31 X IO-4 cm/s from Table 4.5. The arithmetic mean effective porosity, ne, 

is 0.21 from Table 4.4. Therefore, 

..' U = VJnt - ~K~-/ne = 3.31X10_4cm/s X 0.01/0.21 

• '-.-= 1.58X10_5cm/s. 

[End of Example 4.1] 
4-12 Radiological Assessment 

The difficulty associated with the solution of the flow and transport in the 

unsaturated zones leads naturally to approximation methods. The time of 

travel can be estimated by assuming that the mean downward velocity v is 

proportional to the rate of recharge of water at the surface, r, and inversely 

proportional to the mean volumetric water content, 0: 

v = r/8. (4.9) 

The recharge rate, r, can be estimated by methods described in Sect. 4.3.4. 

The volumetric water content, 6, can be conservatively assumed to be equal to 

the field capacity, which is the maximum water content where moisture can no 

longer be held against gravity. Field capacity is equal to the specific retention 

Sn which is defined as 

Sr — ne — n, (4.10) 

where n is the total porosity and ne is the effective porosity. Representative 

values of ne and n are tabulated in Sect. 4.4.2. . , , 

4.3.2 Mass Transport 

The most general form of the mass transport equation is for transport in 

saturated-unsaturated media. If local equilibrium of mass transfer and firstorder chemical reactions are assumed, sorption can ibe represented as a linear 

relationship, and the general mass transport equation can be written as (ANS 

1980) 

RdO^-VA6DVc) + V-(Kc) • + dt R ^ + XOR,, c = 0 , (4.11) 

where 

£ = the concentration of dissolved constituent (g/cm3), 

D = the dispersion tensor (cm2/s), 

V = the flux (cm/s), 

A. = the radioactive decay constant: 

X = ln2/half-life of isotope (1/s). 

The term Rd is the retardation coefficient 

Rd = 1 Kd > 

nt n. 

(4.12) 

(4.13) 

where 

n = the total porosity, 

ne = the effective porosity, 

pb = the bulk density (g/cm3), 

Kd = the distribution coefficient (mL/g). 

Radionuclides in Groundwater 4-13 

More conservatively, by assuming n = ne, Rd can be estimated as 

Rd=l+^Kd. <4-

14> 

ne . 

"An equivalent retardation factor may be defined for fracture flow where the 

exposed area of the fracture is used rather than the porosity (Freeze and 

Cherry 1978). 

Example 4.2. Calculate the retardation coefficient, Rd, for strontium in an 

"average" fine sandstone with a bulk density, pj, of 2.8 g/cm3 and a 

distribution coefficient of 20 mL/g. 

The arithmetic mean values of n, ne are found from Tables 4.3 and 4.4 to 

be 0.34 and 0.21 respectively. The retardation coefficient, Rd, calculated from 

Eq. 4.13 is therefore -• 

Rd = - ^ + ^rX2 0 = 268.3 

0.21 0.21 

Equation 4.14 gives 

Rd = 1 + 

2.8 

0.21 

X20 = 267.7 

[End of Example 4.2] 

For the important case when the medium is assumed to be fully saturated, 

the mass transport equation becomes 

Rd^f-VAD-Vc) + V' 

dt 

Vc 

+ \RdC = 0 . (4.15) 

If the dispersion • tensor is assumed to be homogeneous and isotropic and the 

flux is assumed to be parallel to the x-axis, Eq. 4.15 can be written as 

Rd—-V{DVc) + — Vc + \Rdc = 0 

dt n 

(4.16) 

When the fluid flux is assumed to be uniform and steady, Eq. 4.16 becomes 

dc Dx d2C Dy d2 Dz d2 U dc T-S—2—^- 2 o ^T + ^ ^ + Xc = 0 , (4.17) 

dt Rd dx2 Rd dy2 Rd dz2 Rd dx 

where U is the pore velocity defined by Eq. 4.8a and Dx, Dy, and Dz are the 

dispersion coefficients in the.,*, y, and z directions respectively (cm2/s), as 

described in Sect. 4.4.1. 
4-14 Radiological Assessment Radionuclides in Groundwater 4-15 

The approximate rate of movement of the radionuclide is U/Rd, which 

may be used to estimate the travel time. 

The above equations are strictly valid only for isotropic media (i.e., media 

whose hydraulic conductivity is uniform in all directions) but may be applied 

to slightly anisotropic formations when the dispersivities are obtained from 

field studies. 

4.3.3 Chain Decay of Radionuclides 

Radionuclides decay either to stable products or to another radioactive 

species called a daughter. In some species several daughter products may be 

produced before the parent species decays to a stable element. This process is 

particularly important for modeling actinides and transuranics. In considering 

this process over the transport path of radionuclides, one transport equation 

must be written for each original species and each daughter product to yield 

the concentration of each radionuclide (original species and daughter products) 

at points of interest along the flow path. In a constant one-dimensional velocity 

field, the general equations can be written as (Burkholder and Rosinger 1980) 

dci dc\ 

*«ir+v-r7-D> 

_ dc2 , dc-, 

R

"

2

 i r + u~y = D> 

at dx 

dc, dc, R

--ST

+U-A-D

' 

d

2

cv 

dx2 

d

2

c2 

dx2 

d

2

c, 

dx2' 

— Rdihci . 

~ Rd2*2C2 + RdlhCl . 

I 

"7 Rdlhcl + RdJ-lh-lCi-t 

(4.18) 

where 

Rdl = the retardation factor for species i, 

U = the pore velocity = Vx/ne, 

c( = the concentration of species i, 

Dx = the dispersion coefficient, 

X/ = the decay coefficient for species i. , ' 

Equation 4.18 describes the material balances of the /th member of a decay 

chain and all preceding chain members. 

Analytical models incorporating • hain decay with different sorption 

properties for each daughter are ayailable for up to a three-component chain 

(Burkholder and Rossinger 1980). A simpler analytical formulation applies if 

all daughters are assumed to have equal sorption properties. The concentration 

Cj of the rth daughter in terms of the parent concentration is 

ct = A

l m-l y- i 

e '• 

IK**-1*;)' 

k*j y •• 

(4.19) 

; i 

For long chain decays with sorption considerations, numerical solutions are 

practically mandatory (Burkholder and Rosinger 1980; Dillon et al. 1979). 

,4.3.4, Percolation of Water into the Ground 

An important part of the analysis of the migration of contaminants in 

groundwater is the determination of the rate of release of the contaminant at 

the source (e.g.,, leaching of low-level waste) and determining the speed of 

transport of the groundwater. Both of these aspects of the migration problem 

frequently involve knowing the rate at which water infiltrates the ground either 

from a surface water body such as a river or pond or directly from rainfall 

(percolation). Infiltration is most important for shallow land burial but less 

important for a deep repository not affected by local recharge. For example, 

the source of radioactive contamination at a low-level waste site may be 

limited by the amount of rainfall that penetrates the land surface and comes in 

contact with the buried waste. The flow of groundwater in the water table 

aquifer is directly related to the rate at which surface water recharges it. This 

rate of infiltration of rainwater may be an important boundary condition for 

shallow land burial problems (Aikens et al. 1979). 

Percolation of rainwater is frequently estimated by calculating the water 

budget for the root zone. Wate r enters the root zone through infiltration from 

rainfall and is removed by the evaporation directly from the surface, by 

transpiration from vegetation, and by seepage vertically to the water table. 

Both rigorous (Gupta et al. 1978) and empirical (Thornthwaite et al. 1957) 

methods of performing a water budget are in common use. These methods can 

be found in standard hydrology textbooks along with coefficients that apply to a 

variety of soil and vegetation types and climates (Chow 1964). 

4.4 PARAMETERS FOR TRANSPORT 

AND FLOW EQUATIONS 

4.4.1 Dispersion and Diffusion in Porous Media 

4.4.1.1 Molecular Diffusion 

Dispersion in Eq. 4.11 is actually a combination of molecular diffusion and 

mechanical dispersion, which are processes that irreversibly distribute dissolved 

constituents within porous media. Molecular diffusion results from the random 

movement of molecules at a very small scale. Diffusion within fluids depends on 

fluid properties such as temperature, concentration, and viscosity as well as 

temperature and concentration gradients. In a one-dimensional, nonflowing 

diffusion process, transport due to diffusion is usually related to Fick's law: 

• i : , i£ = -LD,l£ t (4.20) 

dt dx dx 
4-16 Radiological Assessment Radionuclides in Groundwater 4-17 

where D' is the effective diffusion coefficient for porous media, which typically 

varies from about IO- 5 tp IO- 8 cm2/s. The effective diffusion coefficient £>' 

will be lower than the molecular diffusion coefficient in a free liquid because 

diffusion will be inhibited by the pore structure (Evanson and Dettinger 1980). 

4.4.1.2 Dispersion 

Dispersion describes the mechanical mixing of dissolved constituents by 

the complex flow paths the fluid must, take in the porous medium. Variability 

of path length and velocity from the mean results in longitudinal and lateral 

spreading of the dissolved constituents. ,' 

Laboratory investigations have shdwn that in porous media, longitudinal 

dispersion is related to the seepage velocity. For an isotropic medium, the 

dispersion coefficients DtJ can be described in terms of the longitudinal and 

transverse dispersivity (Scheidegger 1961): 

where 

0D,j = a rK«y'+.(ojr-«r^,FyK, (4.21) 

bu = 1 for i = ;, by = 0 for i * ; (Kronecker delta function), 

6 = the volumetric water content, 

aT — the transverse dispersivity (cm), 

• ' • ' ' ,' • 

a, = the longitudinal dispersivity (cm), * !,' 

V = the magnitude of the flux (cm/s), • 

Vh Vj = the components of the flux (cm/s). 

Even in small-scale laboratory experiments in, uniform porous media, 

dispersion processes usually dominate the diffusion ' processes'. Dispersion 

depends on flow, however. For very low flow rates, molecular diffusion, which 

is independent of flow, may dominate the diffusion^ 

4.4.1.3 Macrodispersion 

Experiments with packed laboratory columns generally yield dispersivities 

having dimensions on the order of the median grain diameter, ranging from, 

millimeters to centimeters. If the dispersivities measured in the laboratory were 

used in a transport model for a large aquifer, dispersion would be grossly 

underpredicted. At the aquifer scale, it appears that the heterogeneities in 

permeability, fracturing, stratification and other properties of the medium, 

sampling errors and model approximations, are more important to producing 

dispersive behavior than mixing around individual grains and pores in the 

laboratory-scale experiments (Evenson and Dettinger 1980; Anderson 1979). 

Field studies tend to support the hypothesis that macrodispersion is largely 

a result of heterogeneity in hydraulic conductivity. Dispersivity also apparently 

increases with the length scale of the experiment. There is a tendency for large 

dispersivities to coincide with experiments involving large distances. 

The Fickian analogy for dispersion (Eq. 4.20) does not always behave 

satisfactorily, and the dispersion cannot be characterized with a parameter as 

simple as the dispersivity. The assumptions of homogeneity of the medium 

break down if the heterogeneities are not random or if they are large in 

comparison to the aquifer being modeled (Winograde and Pearson 1976). 

The cases in which the simple dispersion models are likely to fail are 

1. Media in which a few extensive conductivity variations dominate the 

transport process. 

2. Media in which conductivity variations are abrupt, severe, and tend to 

fbltow well-defined paths. 

3. Observations that are made on a scale that is small compared to the scale 

of the variation. 

4. Media that show variations in conductivity that cannot be modeled as a 

random field with apparently random values, spatial extents, and orientation 

assumed by the aquifer properties (Evenson et al. 1980). 

These phenomena have been described generically as "channeling." 

"Evenson et al. (1980) suggest that media in which these phenomena are likely 

to occur (fracture systems, karst, etc.) should not be modeled according to 

Fick's law without extensive justification. Models capable of dealing with these 

problems are extremely complex and beyond the scopie of this book (Evenson et 

al. 1980). 

4.4.1.4 Determination of Dispersion 

It, is frequently the case that the only way the values of dispersion 

coefficients can be determined for a given site is by direct observation of either 

man-made or naturally occurring tracers. Tracers,that have been deliberately 

introduced are used in groundwater studies in single- or double-well pumping 

experiments over relatively short distances and times. Direct tracer methods 

have several disadvantages in groundwater studies: 

1. Because groundwater velocities are rarely large under natural conditions, 

undesirably long times are normally required for tracers to move significant 

distances through the1 ,flow system. For this reason, only small, 

nonrepresentative portions of the flow field can be measured. 

2. Because geological materials are typically quite heterogeneous, numerous 

, observations are usually required to adequately monitor the passage of the 

tracer through the portion of the flow field under investigation. The 

measurements themselves may actually disturb the flow field significantly. 
4-18 Radiological Assessment 

Values of dispersivities obtained from a wide range of tracer experiments and 

also those based on numerical models of observed groundwater solute transport 

cases are presented in Tables 4.1 and 4.2, respectively. (Anderson 1979; 

Evenson and Dettinger 1980). These values represent site-specific cases and 

should be extrapolated to other cases only with extreme caution. Furthermore, 

the dispersivities reported in Table 4.2 probably reflect processes, such as 

numerical dispersion, that are inaccuracies of the mathematical model and are 

not measured in nature. 

4.4.2 Porosity and Effective Porosity' 

The parameters porosity and effective pordsity (or specific yield) are 

necessary for the solution of the flow and solute transport' equations. The 

porosity of a soil or rock is a measure of the interstitial space relative to the 

space occupied by solid material and is expressed quantitatively as the 

percentage of the total volume occupied by the interstices. 

The porosity of a sedimentary deposit depends chiefly oh the shape and 

arrangement of its constituent particles, the degree of assortment of its 

particles, the cementation and compaction to which it has been subjected, the 

dissolution of mineral matter by Water, and the fracturing resulting in open 

joints other than interstices. The porosity of many sedimentary deposits is 

increased by the irregular angular .shapes of its grains. Porosity decreases with 

increases in the variety of size of grains because small grains fill interstices 

between larger grains. Table 4.3 gives representative values of porosity for a 

wide range of soils and rocks. 

The effective porosity is the portion of the porosity that can be considered 

to be available for the flow of groundwater through a porous medium. Not all 

of the water in the interstices of saturated rock or soil is available for flow. 

Part of the water is retained in the interstices by the forces of molecular 

attraction or is trapped in dead-end pores. The amount of water trapped is 

greatest in media having small interstices. Table 4.4 gives representative values 

of effective porosity for a wide range of soils and rocks. 

4.4.3 Hydraulic Conductivity for Saturated Flow 

The hydraulic conductivity, K, for an isotropic, homogeneous saturated 

medium determines the rate at which water moves through the porous medium 

for a given hydraulic gradient. Hydraulic conductivity is a property that 

depends on the properties of both the fluid and the medium and has units of 

velocity (cm/s). A measure of the hydraulic conductivity, which is a property of 

the porous medium alone, is the intrinsic permeability k, that has units of 

length squared and is usually expressed in darcys (one darcy = 9.87 X IO-9 

cm2). 

Radionuclides in Groundwater 4-19 

Table 4.1. Dispersivity values atand aT obtained directly through measurements 

of tracer breakthrough curves in groundwater solute transport 

• 

Setting . 

Chalk River, Ontario 

alluvial aquifer 

Chalk River, strata of 

high velocity 

Alluvial aquifer 

Alluvial, strata of. 

high velocity 

Lyons, France 

alluvial aquifer 

Lyons (full aquifer) 

Lyons (full aquifer) 

' Lyons (full aquifer) 

Lyons (full aquifer) 

Lyons (full aquifer) 

Alsace, France 

alluvial sediments 

Carlsbad, N. Mex. 

fractured dolomite. 

Savannah River, S.C. 

fractured schistgneiss 

Bestow, Calif. 

alluvial sediments 

Dorset, England 

chalk (fractured) 

(intact) 

Berkeley, Calif. 

sand/gravel 

Mississippi limestone 

NTS, carbonate 

aquifer 

Pensacola, Fla. 

'' • limestone 

« i 

(m) 

0.034, 

0.034^-0.1 

,0.5 

0.1 

• , 

0.1-0.5 

, ' 

5 

, 12.0 

.8 

|

 5 . 

7 ' , 

12 

38.1 

134.1 

15.2 

3.1 

1.0 

2-3 

11.6 

15 

10 

«r 

(m) 

3.1-14 

Q.015-1 

0.145-14.5 

0.009-1 

4 . 

Ax" 

(m) 

, 

i 

38.1, 

538 

6.4 

» 00 oo 312 

. o* 

(m/d) 

7.2 

9.6 

13 

9 

0.15 

0.4 

Method 

Single-well tracer 

Single-well 

Two-well 

Two-well 

Single-well 

Single-well 

Single-well test 

with resistivity 

Single-well test 

with resistivity 

Single-well test 

with resistivity 

Single-well test 

with resistivity 

Environmental 

Two-well tracer 

Two-well 

Two-well 

Two-well 

Two-well 

311-1382 Multiwell trace 

06 

test 

Single-well 

Two-well tracer 

Two-well 

aAx' = distance between wells in two-well test. 

bD = groundwater seepage velocity. . 

Source: Evenson, D. E., and Dettinger, M. D. 1980. Dispersive Processes in Models 

of Regional Radionuclide Migration, University of California, Lawrence L.vermore 

1

 Laboratory, Livermore. 
I. 

4-20 Radiological Assessment Radionuclides in Groundwater 4-21 

Table 4.2. Dispersivity values aL and. aT obtained by calibration of numerical 

transport models against,observed groundwater solute transport 

Setting 

Rocky Mtn. Arsenal 

alluvial sediments 

Arkansas River Valley 

coalluvial sediments 

California 

alluvial sediments 

Long Island 

glacial deposits 

Brunswick, Ga. 

limestone 

Snake River, Idaho 

fractured basalt 

Idaho, fractured 

basalt 

Hanford site, Wash. 

fractured basalt 

Barstow, Calif. 

alluvial deposits 

Roswell Basin, N. Mex. 

limestone 

Idaho Falls, lava 

flows and sediments 

Barstow, Calif. 

. alluvial sediments 

Alsace, France 

alluvial sediments 

Florida (SE) 

limestone 

Sutter Basin, Calif. 

alluvial sediments 

«£ 

(m) 

30.5 

30.5 

30.5 

21.3 

61 

91 

• 

91 

30.5 

• 

61 

21.3 

91 

61 

15 

6-7 ;, 

80-2Q0 

Olf 

, (m) 

, 30.5 

, 9.1 

9.1 

4.3 

20 

136.5 

91 

18 

18 

137 

0.18 

1 

0.7 

8-20 

: Axa 

(m) 

305 

660X1320 

305 , y 

Variable 

(50-300) 

Variable 

640 

640 

305 

Variable 

3X152 

1

' 

Variable 

(2-20 km) 

1 

• , ,JS Method' 

(m/d) 

Areal (moc) 

Areal (moc) 

• • 

Areal 

0.4 Areal (fe) 

Areal (moc) 

Areal 

Areal (fe) 

Areal (rw) 

Areal (fe) 

Areal 

Areal 

Profile (fe) 

Profile 

Profile 

3-D(fe) 

"Ax = grid size in program. 

bU = groundwater seepage velocity. ' , 

c

(fe) indicates use of a finite-element model; (moc) indicates method of characteristics; and (rw) indicates a random-walk model. 

Source: Evenson, D. E., and Dettinger, M. D. 1980. Dispersive Processes in Models 

of Regional Radionuclide Migration,' University of California, Lawrence Livermore 

Laboratory, Livermore. 

Table 4.3. Typical values of porosity of aquifer materials 

Aquifer material 

Igneous Rocks ' 

Weathered granite 

Weathered gabbro 

Basalt 

Sedimentary Materials 

Sandstone 

1 1

 Siltstone 

Sand (fine) 

Sand (coarse) 

Gravel (fine) 

Gravel (coarse) 

Silt 

Clay 

Limestone 

Metamorphic Rocks 

Schist 

Number of 

analyses 

8 

4 

94 

65 

7 

245 

26 

38 • 

15 

281 

74 

74 

18 

. Range 

0.34-0.57 

0.42-O.45 

, 0.03-0.35 

0.14-0.49 

0.21-0.41 

0.25-0.53 

0.31-0.46 

0.25-0.38 

0.24-0.36 

0.34-0.51 

0.34-0.57 

0.07-0.56 

0.04-0.49 

Arithmetic 

mean 

0.45 

0.43 

0.17 

0.34 

0.35 

0.43 

0.39 

0.34 

0.28 

0.45 

0.42 

0.30 

0.38 

, Source: McWhorter, D. B., and Sunada, D. K. 1977. 

Ground-Water Hydrology and Hydraulics, Water Resources 

Publications, Fort Collins, Colo. Reprinted with permission. 

Hydraulic conductivity, K, and intrinsic permeability, k, are generall; 

related by the equation 

r>-

(4.22 

where g is the acceleration of gravity, p is the density of the fluid, and p. is th 

viscosity of the fluid. Table 4.5 gives representative values of hydrauli 

conductivity in centimeters per second for a sample of common poroui 

jnaterials (McWhorter and Sunada 1977). 

Environmental factors may affect the hydraulic conductivity of a givei 

porous medium. For example, ion exchange on clay and colloid surfaces wi] 

cause changes in mineral volume and pore size and shape. Changes in pressun 

may cause compaction of the material or may cause gases to come out o 

solution, which would reduce the hydraulic conductivity (Davis and De Wies 

i965). 

4.4.4 Adsorption and Retardation Coefficients 

An important mechanism in retarding the migration of radionuclides i 

groundwater is sorption, which is defined to include all rock-water interaction-
4-22 Radiological Assessment 

Table 4.4. Typical values of effective porosity 

(or specific yield) of aquifer materials 

Aquifer material 

Sedimentary Materials 

Sandstone (fine) 

Sandstone (medium) 

Siltstone 

Sand (fine) 

Sand (medium) 

Sand (coarse) 

Gravel (fine) 

Gravel (medium) 

Gravel (coarse) 

SUt 

Clay 

Limestone 

Wind-Laid Materials 

Loess 

Eolian Sand 

Tuff 

Metamorphic Rock 

Schist 

Number of 

analyses 

47 

10 

13 

287 

297 

143 

33 

13 

9 

1

 299 

27 

32 

1 

5 " 

14 

90 

11 

Range 

0.02-0.40 

0.12-0.41 

0.01-0.33 

0.01-0.46 

0.16-0.46 

0.18-0.43 

0.13-&40 

0.17-0.44 

0.1,3-0.25 

0.01-0.39 

,0.01-0.18 

--0-0.36 

, 

0.14-0.22 

0.32-0'47 

0.02-0.47 

0.22^.33 

Arithmetic 

mean 

0.21 

0.27 

0.12 

0.33 

0.32 

0.30 

0.28 

0.24 

0.21 

0.20 

0.06 

0.14 

• 

0.18 

0.38 

0.21 

' 

0.26 

Source: McWhorter, Di B., and Sunada, D- K. 1977. 

Ground-Water Hydrology and Hydraulics, Water Resources 

Publications, Fort Collins, Colo. Reprinted with permission. 

that cause the radionuclides to migrate at a slower rate than the groundwater 

itself. The amount of sorption is dependent on both the chemistry of the water 

and of the rocks; and, because some of the chemical reactions are slow, it is a 

function of time as well. 

Values of sorption coefficients are required to calculate the travel time of 

key radionuclides from the source to the biosphere. The sorption coefficients 

are usually obtained using a standard batch test where rocks are put in contact 

with groundwater in which small amounts of radionuclides have been mixed. 

The problem with this type of approach is that more detailed geochemical data 

are necessary to support the validity of the sorption measurement over the 

expected travel time of the radionuclides (which may be of the order of 

thousands of years). To provide the justification for using simple sorption 

coefficients, a detailed understanding of the geochemical mechanisms pf rockwater interactions must be attained. Such mechanisms as dissolution/ 

precipitation, complexing, adsorptibn/desorption, phase transformations, and 

solubility should be understood for radionuclides of interest in the geochemical 

Radionuclides in Groundwater 4-23 

Table 4.5. Typical values of hydraulic conductivity of porous materials 

Material 

Number of 

analyses 

Range 

(cm/s) 

Arithmetic 

mean 

(cm/s) 

Igneous rocks 

Weathered granite 

Weathered gabbro 

Basalt 

Sedimentary materials 

Sandstone (fine) 

Siltstone 

Sand (fine) 

Sand (medium) 

Sand (coarse) 

Gravel 

SUt 

Clay 

Metamorphic rocks 

Schist 

7 

4 

93 

20 

8 

159 

255 

158 

40 

39 

19 

17 

(3.3-52) X IO-" 

(0.5-3.8) X IO-4 

(0.2-4250) X 10- 8 

(0.5-2270) X 10"6 

(0.1-142) X 10-8 

(0.2-189) X IO-4 

(0.9-567) X IO"4 

(0.3-6610) X IO-4 

(0.3-31.2) X 10-1 

(0.09-7090) X 10-7 

(0.1-47) X 10-8 

(0.002-1130) X IO-6 

1.65 X 10-3 

1.89 X IO-4 

9.45 X 10-6 

3.31 X IO-4 

1.9 X 10~7 

2.88 X 10-3 

1.42 X 10-2 

5.20 X 10"2 

4.03 X 10"' 

2.83 X 10"5 

9 X 10"8 

1.9 X IO"4 

Source: McWhorter, D. B., and Sunada, D. K. 1977. Ground-Water Hydrology and 

Hydraulics, Water Resources Publications, Fort Collins, Colo. Reprinted with 

permission. 

environment. The effect of heat, radiation, or high concentrations of chemicals 

will be particularly important close to the source of release in some situations. 

Much of this understanding for shorter periods of time and close to the points 

of release can be obtained through a combination of laboratory and field experiments combined with data from natural systems that can be used as analogs. 

However, over longer time periods or far from the points of release, all of the 

data must be obtained from studies of the natural system. 

Natural analogs of interest for application to radionuclide migration 

include hydrothermal ore deposits, intrusive magmas into generic host rocks, 

uranium ore bodies, rich thorium deposits, natural fission reactors, and 

underground nuclear explosions (Klingsberg and Duguid 1979). Also, the 

behavior of natural radionuclides and their decay products in host rock 

formations can provide the data necessary to choose conservative sorption coefficients for use in the transport models are conservative over the range of geochemical conditions and the transport travel times expected. 

Tables 4.6 and 4.7 give typical ranges of distribution coefficients (Kd) for 

several significant', randionuclides in an assortment of rocks and soils 

(Isherwood 1981). These tables illustrate some of the sensitivity of Kd to 
4-24 Radiological Assessment Radionuclides in Groundwater 4-25 

Table 4.6. Distribution coefficients: strontium and cesium 

Kd{mL/g) 

Table 4.7. Distribution coefficients: thorium 

and uranium 

' 

Basalt, 32-80 mesh 

Basalt, 0.5-4 mm, 300 ppm TDS 

Basalt—0.5-4 mm, sea water 

Basalt-fractured in situ measurement 

Sand, quartz—pH 7.7 

Sands 

Carbonate, greater than 4 mm 

Dolomite, 4000 ppm TDS 

Granite, greater than 4 mm 

Granodiorite, 100-200 mesh 

Granodiorite, 0.5-1 mm 

Hanford sediments 

Tuff 

Soils 

Shaley siltstone greater than 4 mm 

Shaley siltstone greater than. 4 mm 

Alluvium, 0.5-4 mm 

Salt, greater than 4 mm 

saturated brine 

"Sr 

i 

16-135 . 

220-1220 

1.1 

3 ' ' •' 

1.7-3^8 

13-43 

0.19 

5-14 

1/7 J ' 

4-9 

11-23 

50 

45-4000 

19-282 

8 

1.4 

48-2454 

0.19 

Cs 

792-9520 

39-280 

6.5 

•• 

22-314 

100 

13.5 

34.3 

8-9 

1030-1810 

300 

800-17800 

189-1053 

309 

102 

121-3165 

0.027 

Source: Isherwood, D. 1981. Geoscience Data Base Handbook 

for Modeling a Nuclear, Wdste Repository, NUREG/CR-0912, vols. 

1 and 2, U.S. Nuclear Regulatory Commission. 

factors such as particle size and chemistry of the water phase. Values of Kd 

should be extrapolated to situations other than those for which they were1 

determined only with extreme caution. 

4.5 METHODS OF SOLUTION FOR 

GROUNDWATER MOVEMENT 

4.5.1 Introduction 

Over the past several years,1 numerous mathematical models have been 

developed to simulate the flow of groundwater and the transport of radioactive 

and chemical substances, particularly in, the field df waste management. 

Discussion of the virtually hundreds of groundwater models is beyond the scope 

of this chapter, but two excellent compilations of groundwater models are 

available (Bredehoeft 1978; SAI 1981). , 

Groundwater mathematical models can be broadly classified as either 

numerical or analytical. Numerical techniques are usually direct solutions of 

AT, (mL/g) 

• " ' • ' . 

160,000 

400,000 

160,000 

40-130 

310-470 

270-10,000 

8 

60 

120 

1000 

< 100,000 

62,000 

4400 

300 

2000 

270 

4.5 

2.9 

Conditions 

Thorium 

Silt loam, Ca-saturated clay, pH 6.5 

Montmorillonite, Ca-saturated clay, pH 6.5 

Clay soil, 5 mM Ca(N03)2, pH 6.5 

Medium sand, pH 8.15 

Very fine sand, pH 8.15 

Silt/clay, pH 8.15 

Schist soil, 1 g/L Th, pH 3.2 

Schist soil, 0.1 g/L Th,"pH 3.2 

Illite, 1 g/L Th, pH 3.2 

Illite, 0.1 g/LTh, pH 3.2 

Illite, 0.1 g/LTh,pH>6 

Uranium 

Silt loam, U(VI), Ca-saturated, pH 6.5 

Clay soil, U(VI), 5 mM Ca(N03)2, pH 6.5 

Clay soil, 1 ppm UO+2, pH 5.5 

Clay soil, 1 ppm UO+2, pH 10 

Clay soil, 1 ppm UO+2, pH 12 

Dolomite, 100-325 mesh, brine, pH 6.9 

Limestone, 100-170 mesh, brine, pH 6.9 

Source: Isherwood, D. 1981. Geoscience Data Base 

Handbook for Modeling a Nuclear Waste Repository, 

NUREG/CR-0912, vols. 1 and 2, U.S. Nuclear 

Regulatory Commission. 

the differential equations describing water movement and solute transport, 

using methpds such as finite differences or finite elements. These methods 

always require a digital computer, a large quantity of data, and an experienced 

modeler-hydrologist. The validity of the results from numerical models depends 

strongly on the quality and quantity of the input parameters. Analytical 

models are usually approximate or exact solutions to simplified forms of the 

differential equations for water movement and solute transport. Such models 

are simpler to use than numerical models and can generally be solved with the 

aid of a calculator, although computers are also used. Analytical models are 

much more severely limited to simplified representations of the physical 

situations. However,, they are extremely useful for scoping the problem to 

determine data needs or the applicability of more detailed numerical models. 

Several of the more important types of numerical and analytical models 

are discussed below. 
4-26 Radiological Assessment Radionuclides in Groundwater 4-27 

4.5.2 Numerical Methods 

4.5.2.1 Finite Difference 

One approach that has beep applied to the solution of groundwater 

equations involves finite-difference approximations. To apply these 

approximations, the region under consideration is usually divided into a 

rectangular grid. The intersections of the grid are called nodal points and 

represent the position at which the solution for unknown values such as 

hydraulic head are obtained. When difference equations are written for all 

nodes and boundary conditions are applied, a system of n algebraic equations in 

« variables can be solved for the variables at each node for each time increment 

(Faust and Mercer 1980). ' 

4.5.2.2 Finite Element 

The finite-element method is a numerical method where a region is 

divided into subregions, called elements, whose shapes are determined by a set 

of points called nodes (similar to the finite-difference grid). The first step is to 

derive an integral representation of the partial differential equations. This is 

commonly done by using the method of weighted residuals or the variational 

method. The next step is tp approximate the dependent variables (head or 

concentration) in terms of interpolation fuhctibns called basis functions. 

Once the basis functions are specified and the elements defined, the 

integral relationship must be expressed for each elernenf as a function of the 

coordinates of all nodal points of the element; Then' the values of the integrals 

are calculated for each element. The values'for all elements are combined and 

boundary conditions applied to, yield a system of first-order linear differential 

equations in time (Faust and Mercer 1980). 

4.5.2.3 Method of Characteristics 

The method of characteristics is used in convection-dominated transport 

problems where finite-difference and finite-element: approaches suffer from 

"numerical dispersion" or solutions that oscillate. The approach is not to solve 

the transport equations directly but to solve an equivalent system of ordinary , 

differential equations that are obtained by rewriting the transport equation 

using the fluid particles as a reference point. This is accomplished numerically 

by introducing a set of moving points (reference particles) that can be traced 

within the stationary coordinates of a finite-difference grid block and allowed 

to move a distance proportional ,to the velocity and elapsed time. The moving 

particles simulate the convective transport because concentration is a function 

of spreading or convecting of the particles. Once the convective effects are 

known, the remaining parts of the transport equation are solved using finitedifference approximations (Faust'and Mercer 1980). 

4.5.2.4 Random-Walk Method 

A method similar in many ways to the method of characteristics is the 

random-walk method. In this approach, a particle-tracing advection model is 

used to simulate advection. At the end of each advection time step, the 

particles are dispersed by being displaced a random distance in a random 

direction. Concentrations are calculated by counting the resulting number of 

elements in each cell and comparing this to the initial conditions. This solution 

technique is based on the realization that a normal probability distribution is a 

solution to Fick's Law of'diffusion. This method of transport modeling is easily 

implemented and provides simulations whose accuracy are limited only by the 

number of particles that can be traced: The main disadvantage of this method 

and the method of characteristics is the difficulty and expense of keeping track 

of large numbers of particles (Evenson. and Dettinger 1980). 

, I I 

4.5.2.5 Flow Network Models 

The numerical simulation by finite differences or finite elements of 

groundwater flow and solute transport problems in two and three dimensions 

can be costly in terms of computational resources. Flow network models such 

as the network flow andi transport model (NWFT) (Campbell et al. 1980) are 

those that can be used to describe two- or three-dimensional fields in a much 

more efficient way by a network of interconnecting one-dimensional flow 

segments. Fluid discharge and velocity are determined by requiring 

conservation of mass at the segment junctions. Radionuclide migration from 

the, points of release is calculated by assuming that transport occurs along a 

single one-dimensional path having a length equal to the total migration path 

length. The network model is particularly useful when it is used in conjunction 

with a more complicated two- or three-dimensional model to first define the 

flow, and concentration field for a particular example. The network model is 

first matched or tuned to the results of the complicated model. The tuned 

network model may then be used for further computations with a much 

smaller commitment of computational resources then the original model for 

further runs and sensitivity experiments. 

4.5.2.6 Advection Models 

There are groundwater solute modeling situations where the phenomenon 

' of dispersion, together wtyh its many uncertainties, is only a minor factor in 

describing the transport of contaminants in groundwater. For example, the flux 

of contaminant entering a river that is recharged from a contaminated aquifer 

is much less sensitive to dispersion than the concentration in a particular well. 

In the former case, the contaminated groundwater would enter over a wide 

area, which would tend to smear out the effect of dispersion. For similar 

Reasons, the transport from nonpoint sources of contamination such as large 
4-28 Radiological Assessment 

low-level radioactive waste landfills would diminish the sensitivity of modeled 

results to dispersion. A flow model can be used to generate a velocity potential 

field and stream lines. The flow patterns from the.sources to the sinks can then 

be used to formulate the arrival time distribution, which can be used to 

calculate the concentration of flux of a contaminant at the point of use. Either 

numerical or analytical solutions of the flow equations are used to estimate 

groundwater velocities, the length of the path of a contaminant, and the arrival 

time distribution (Nelson 1978). 

4.53 Analytical Models 

Analytical groundwater transport models can be used for certain types of 

analyses where available data do not warrant, a more complicated study. Such 

models are useful for scoping the transport problem and may frequently be 

adequate for regulatory needs if model coefficients are'ch,6seh conservatively. , 

A series of simple analytical models that have' been used at the U.S. 

Nuclear Regulatory Commission (NRC) is presented below. Many of. these 

models have been computerized and are available from the NRC (Codell et al. 

1982). In their simplest forms, however, they may be used with the aid of only 

a calculator. 

The models are developed for the limiting case of undirectional saturated 

convective transport of a single dissolved substance with three-dimensional 

dispersion in an isotropic aquifer as discussed in Sect. 4.3.2: 

A. d2

dc j U dc _ c 

dt Rd dx Rd dx2 ' Rd dy2 ' Rd dz + 2 

\ d2c , Dz d2c , 

r -r~ ~^ r ~ Ac , 

(4.23) 

where 

c is the concentration in the liquid phase (Ci/cnr), 

Dx, Dy, D2 are the dispersion coefficients in the x, y, and z directions 

respectively (cm2/s), 

\ is the decay coefficient = In 2/half-life (1/s), 

U is the x component groundwater pore velocity (cm/s), 

Rd is the retardation coefficient (dimensionless). 

The dispersion coefficient can be approximated from Eq. 4.21. In this case 

y2 = V-i = 0, V\ = V, and 9 can be approximated for saturated flow by the 

effective porosity, ne. Also, since U = V/ne, 

Dx = aLU 

Dy = aTU 

,DZ = aTU , 

(4.23a) 

(4.23b) 

(4.23c) 

where aL and aT are the longitudinal and transverse dispersivities respectively. 

Radionuclides in Groundwater 4-29 

4.5.3.1 Point Concentration Model 

The first model developed is used for calculating the concentration in the 

aquifer at some point downgradient of a release (e.g., water supply well). 

Equation 4.23 is solved in terms of Green's functions: 

ct = 

1 

neR, 

-X(x,t)Y(y,t)Z(z,t), 

(4.24) 

where c, is the concentration at any point in space for an instantaneous onecurie release, rie is the effective porosity of the medium, and X, Y, Z are the 

Green's functions in the x, y, z coordinate directions, respectively. Equation 

4.24 has been developed for a variety of boundary and source configurations: 

1. For the case of a point source at (0, 0, zs) in an aquifer of infinite lateral 

(x, y) extent and depth b, as illustrated in Fig. 4.2, 

d = 

1 

neR, 

where 

X{ = 

y/AitDxt/Rd 

exp 

Yi ** 

-Xi Y\Z\, 

(x-Ut/Rd)

2 

ADxt/Rd 

„2 

- \ t 

tjAltDyt/Rd 

exp 

• 4Dyt/Rd 

Zx = 4- 1 + 2 % exp 

m - l 

m 

XT?Dz

t 

b

2Rd 

cos m ir— cos mir — b b 

(4.25) 

(4.26) 

(4.27) 

(4.28) 

2. For the vertically averaged concentration in case 1 above (equivalent to a 

vertical line source of length b), 

Ci = 

1 

neR, 

eitd 

-XxY\Zi, 

where 

z2 = -r ; 

(4.29) 

(4.30) 

3. For a horizontal line source of length w centered at (0, 0, zs), as illustrated 

in Fig. 4.3, 
4-30 Radiological Assessment 

' ORNL-DWG 82-14406 

POINT SOURCE AT 

x = 0, y = 0, z = z. 

•?>?«-

TOP OF AQUIFER OR 

BOTTOM OF AQUIFER 

Figure 4.2. Idealized groundwater system for point concentration model, point source. 

HORIZONTAL LINE SOURCE 

CENTERED AT 

x = 0, y = 0, z = Zt 

ORNL-DWG 82-14407 

->=^-^ — 

TOPOF AQUIFER OR 

BOTTOM OF AQUIFER 

Figure 4.3. Idealized groundwater system for point concentration model, horizontal line 

source. 

Radionuclides in Groundwater 4-31 

c,- = neR< 

X\Y2Z\, 

where 

Y2 = -L(erf^±^ + erf^^ = 

i-2w \ y/ADyt/Rd -J*Dyt/Rd 

(4.31) 

(4.32) 

>, ^nd erf is the error function. Tables of the error function are available in 

• standard mathematical texts (Abramowitz 1970); 

4. For the vertically averaged concentration in case 3 above (equivalent to an 

area source of width w and depth b), 

1 

ntRd 

X\Y2Z2\ (4.33) 

5. For a point source at (0, 0, zs) in an aquifer of infinite lateral extent and 

depth, 

c,- = 

1 

ntRd 

X\ Y\Z3 > 

where 

Z3 = 

1 

•tfixDj/Ri exp 

(z~zs)

2 

4Dzt/Rd 

+ exp 

(z+z,)2 

4Dzt/Rd 

(4.34) 

(4.35) 

6. For a horizontal line source of width xv centered at (0, 0, zs) in an aquifer 

' of infinite lateral extent and depth, 

Ci = 

1 

neR< 

XXY2Z3', (4.36) 

7. For a horizontal area source of length / and width w centered at (0, 0, 0) in 

an aquifer of constant depth b, as illustrated in Fig. 4.4, Eq. 4.24 becomes 

Cl = 

1 

neR( 

•X2Y2Z2, 

(4.37) 

where 

X, = 

J_1 

2fi 

erf 

X+2 

ut 

Rd 

y/4Dxt/Rd 

-er f 

.8 ut 

Rd 

y/4Dxt/Rd 

exp( — X t). 

(4.38) 
4-32 Radiological Assessment 

ORNL-DWG 82-14408 

SOURCE AREA 

x 

Figure 4.4. VerticaUy averaged groundwater dispersion model. 

Example 4.3. Concentration in an aquifer of limited thickness. 

One curie of a radioactive pollutant leaks quickly into a water table 

aquifer through a highly permeable ground cover over a square surface area 50 

m on a side. The pollutant has a half-life df 30 y. A well tracer test indicates 

that the groundwater is moving in the direction of two wells at a speed, U, of 

1.5 m/d and that the longitudinal and transverse dispersivities, ar. and aT, are 

20 to 10 m, respectively. 

The saturated thickness of the water table aquifer, b, is 50 m and has an 

effective porosity, ne, of 0.2. The pollutant has been determined to have a 

retardation coefficient, Rd, of 20 jn the aquifer. 

Calculate the concentration of the pollutant in wells whose downgradient 

coordinates with respect to the center of the source area are 

Radionuclides in Groundwater 4-33 

(a) x = 200 m, y = 0 m 

(b)x = 400 m, y = 50 m 

The wells are screened over the entire depth of the aquifer. 

Case 7 in Sect. 4.5.3.1 applies to this example, since the source is a 

horizontal area type and the wells are screened over the total depth, which 

would vertically average the concentration. 

Equation 4.37 is therefore evaluated with Green's function: 

A^ determined by Eq. 4.38, 

^determined by Eq. 4.32, and 

Z2 determined by Eq. 4.30, 

The dispersion coefficients are calculated by Eqs. 4.23a and 4.23b. 

Dx = aLU = 20 X 1.5 = 30 m2 

Dy = aTU = 10 X 1.5 = 15 m2 

Figure 4.5 shows the concentration as a function of time calculated for the 

two wells. [End of Example 4.3] 

ORNL-DWG 82C-20751 

15 

10 

OO 

o 

E 

o 

X=200 m, Y=0 

X=400 m, Y=50 m 

I 

i • • 5 . • • 1 0 15 

TIME (1000 d) . 

Figure 4.5. Concentration in downgradient Wells for Example 4.3. 

20 

CO 
4-34 Radiological Assessment 

4.5.3.2 Flux Models 

The flux model is used to calculate the discharge rate of a radionuclide 

entering a surface water body that has intercepted the'aquifer containing the 

transported material as depicted in Fig. 4.6. It'is assumed that all material 

ORNL-DWG 82-14409 

SOURCE AT x = 0 

APPROXIMATE DISTANCE 

TO RIVER i 

< x . » 

^A/AAAAAM. 

SURFACE WATER 

^STREAMLINES ^ 

Figure 4.6. Groundwater'-surface water interface, flux model. 

entering the aquifer eventually enters the surface water except for that which 

has been lost through radioactive1 decay. The assumptions that apply to the'• 

point concentration model also apply to this model. The model provides only 

the rate of input to the surface, water at an average distance x downgradient 

from the surface. Actually, the contamination would enter the surface water as 

a diffuse patch, but the model described here gives no information about the 

spatial distribution of this patch. ' . ," 

In the unidirectional flow field assumed, the flux, F (Ci/s) of material 

crossing an area dA = dy. dz perpendicular to the x axis is described by, the 

equation 

dF 

dA *-*£ ne, 

(4.39) 

where c is the concentration in the dissplved phase. The total flux across, the 

plane would be 

Radionuclides in Groundwater 4-35 

J

. b nCO 

j-r Uc-D, dc_ 

dx 

dydz (4.40) 

4.5.3.3 Source Released from a Vertical Plane (x = 0) 

If Q is the concentration from an instantaneous release of 1 Ci at x = 0 

and time t — 0, as described by Eq. 4.25, then the resulting flux at distance x 

downgradient would be 

F, = 

, Ut\ 

Kd\ 

4y/DxTt3/Rd 

exp 

Ut 

Rd 

4Dxt/Rd 

~\t (4.41) 

4.5.3.4 Horizontal Area Source 

For conditions expressed by Eq. 4.37, the corresponding flux would be 

1 

F, = : [ -j^ rJ*Dxt/Rd [ erf(z,) - erf(z2) ] 

2i-JirDxt/Rd 

-|^iexp(-z 2)-exp(-z 2

2

)]]exp(-XO ,

 (442 ) 

where 

u • . 2 . x u 2. —trt + -z x——t — — 

Rd 2 .. • . Rd 2 

z\ = ~ j===-=-- , and z2 = 

,AJ4Dxt/Rd y/4Dxt/Rd 

Example 4.4. For the same conditions in the previous example (Sect. 

4.5.3.1), calculate, the flux of the pollutant into a river intercepting the 

groundwater flow, which is a distance x of 2000 m downgradient from the 

, center of the source. ' 

Equation 4.42 applies in this case. Figure 4.7 shows the flux into the river 

as a function of time. [End of Example 4.4] 

4.5.3.5 Generalization of Instantaneous Models 

i Equations 4.8. and 4.23 are formulated only in terms of instantaneous 

releases. They . can be generalized for arbitrary releases by use of the 

convolution integral: 

6

 =••••IjM8,(t-T)dT, •'n 

(4.43) 

ro 
4-36 Radiological Assessment 

ORNL-DWG 82C-20750 

O 

3. 

x 

.12 16 20' . 24 ; 

TIME (1000 d) 

,28 32 36 

Figure 4.7. Flux of pollutant into river for Example 4.4/ 

where 0 is the solution at time t for the arbitrary release, 0,(f — T) is the 

solution at time (f — T) for an instantaneous release at (t — T) =T 0, 

and /(T ) is the source release rate at T in curies/s. 

Certain analytical solutions can be found to Eq. 4.43 for simple source 

release rate functions. For example, Wilson develops the solution to Eq. 4.43 for 

a continuous release in terms of, the "well function" (Wilson and Miller 1978). 

Most useful solutions to Eq. 4.43 use numerical integration, generally involving 

a digital computer. 

Several special precautions must be taken, however, to Deserve computational accuracy, because the terms within the integral of Eq. 4.43 can be 

very nearly zero over part of the integration range. Computer programs for 

solving the equations in this section are described by Codell et al. (1982). Program listings in BASIC and FORTRAN are given in this reference. A computer tape of the programs is also available from NRC. 

An alternate method that could be used to simulate a continuous source 

function is to present the continuous source as a series of instantaneous ones. 

The analytical solutions are then linearly summed. Complicated area source 

Radionuclides in Groundwater 4-37 

terms can also be solved in an analogous fashion by representing the source 

area by a series of point sources and linearly summing the solutions. 

4.5.4 Simplified Analytical Methods for Minimum Dilutions 

Simplified forms of the equations of Sect. 4.5.3 have been developed for 

calculating the minimum dilutions (i.e., maximum concentration) of volume VT 

of a substance instantaneously released from a point source into an aquifer. 

4.5.4.1 Dilution at Downgradient Wells in Confined Aquifiers for 

an Instantaneous Point Source at the Surface 

At some distance downgradient from a release at the surface of a confined 

aquifer, the concentration can be considered to be mixed in the vertical 

direction. Close to the point of release, pr in an unconfined aquifer, the vertical 

dispersion will not be influenced by the vertical boundaries of the aquifer. 

Between these regions there is a region where the concentration cannot be 

considered mixed, but the boundaries (top and bottom) affect the dispersion. 

The degree of vertical mixing can be characterized in a confined aquifer of constant thickness and uniform transport properties by the factor 

<*>. = 

b

2 

aTx 

(4.44) 

where 

aj 

'••' :b 

• ' x 

the vertical (transverse) dispersivity, 

the thickness ofthe aquifer (ft), 

the distance downgradient of the release. 

The factor <f> can be used to characterize the aquifer in three approximate 

regions: 

(a) If <t> < 3.3, ,the release may be considered to be within 10% of being 

vertically mixed in the aquifer; 

(b) If 0 > 12, the release may be considered to be within 10% of being 

' unaffected by the vertical boundaries of the aquifer; 

(c) If 3.3 < <t> < 12, the release is neither completely mixed nor unaffected by 

the boundaries. 

Different methods apply to each of the three regions. 

TJ 

ho 
4-38 Radiological Assessment 

Vertically Mixed Region (<f> < 3.3). For an instantaneous release at x = 0, 

the minimum dilution corrected for decay directly dpwngradient of a source 

would be i ' 

DL = Rd4rne '—exp(Xf), 

(4.45) 

where 

DL = minimum dilution =r CQ/C, 

Rd = retardation coefficient* 

ne = effective porosity, 

VT = volume of liquid source term (cm3), 

aL,aT = dispersivities (cm) in the indicated direction, 

x = distance downgradient (cm), 

b = aquifer thickness (cm), 

t = travel time (y), 

X = decay constant = In 2/11/2(1 y). 

The travel time, t, can be approximated as 

" ' - TJR«> 

where U is the pore velocity defined by Eq. 4.8a. 

(4.46) 

Unmixed Region (<t> > 12). For an instantaneous release at x = 0 on the 

surface of the aquifer, the minimum dilution of the surface of the aquifer 

directly downgradient from the source would be 

neRd{.4irx)3r2ylaLa\ 

DL = — exp(A<), 

(4.47) 

where aL, aT are dispersivities in the indicated direction and the other terms 

are as previously defined. 1 

Intermediate Region (3.3 < d> < 12). For an instantaneous release at x = 

0 on the surface of an aquifer, the minimum, dilution on the surface of the 

aquifer directly downgradient from the source would be 

D, = 

Rd4irne y/aiaj-xb 

VTFC*V ', 

exp(Af), 

(4.48) 

Radionuclides in Groundwater 4-39 

where 

. F(<f>) =1+2 2 « P 

n - l 

"-It V (4.49) 

and the other terms are as previously defined. 

, The function F(0) is conveniently plotted in Fig. 4.8. It can be easily seen 

that for small value's of 0, F approaches the value of 1.0, which yields the 

vertically mixed case. For large values of <t>, the slope of F is 1/2, and the 

unmixed case prevails. This method may be used for any value of 4> that can 

be read on Fig. 4.6. 

5 — 

ORNL-DWG 82-14410 

2 — 

10 20 50 100 

aTx 

Figure 4.8. Mixing factor for confined aquifers. 

4.5.4.2 Groundwater-Surface Water Interface—Instantaneous Source 

For an instantaneous release to the groundwater at x =0 , the minimum 

dilution in an intercepting river, corrected for decay, can be found to be 

D, = 

2RdQrJiraLx 

UVT 

exp(Xr) , 

(4.50) 

ro 

ro 
4-40 Radiological Assessment 

where 

Q 

<*L 

U 

flow rate of river (cm3/s), 

the longitudinal dispersivity of the aquifer (cm), 

the volume of release (e.g., tank volume) (cm3), 

pore velocity of groundwater (cm/s). 

4.5.5 Models for Population Doses 

If interdictive or mitigative methods to restrict water use are not taken 

into account, population dose .from contaminated groundwater is proportional 

to the time-averaged concentration. Relatively simple equations can be used for 

estimating average concentration in ground water or in surface water supplies 

contaminated by groundwater. Two equations are presented below that are 

useful for population dose estimates. 

4.5.5.1 Quantity of Released Radioactivity Crossing a Vertical Plane 

In the case of groundwater flow to an intercepting river, the total quantity 

M (curies) of the dissolved substance entering the river would be 

M J

*00 

Fdt, 

o 

(4.51) 

where F is the flux defined for either an instantaneous point or vertical plane 

source by Eq. 4.40 or a horizontal area source by Eq. 4.42. Equation 4.51 can 

be integrated graphically or numerically and in some cases may have an 

analytical solution. :, 

If dispersion is relatively small (e.g., yx « £), the following 

approximation may be used: 

M = M$e w curies (4.52). 

' • i 

where M0 is the quantity of radioactivity released ihstantaneously from the 

source (curies), t is the travel time (y), and X is the decay coefficient (1/y). 

If the substance is being released from the source at a rate proportional to 

the quantity remaining (e.g., a leaching source term), 

M - Mr, X'+X 

e-» (4.53) 

where X' is the release rate from the source (1/y) and M0 is the initial quantity 

of material in the source term (curies). 

Radionuclides in Groundwater 4-41 

4.5.5.2 Direct Groundwater Usage 

A model for calculating the quantity of a radionuclide ingested by a 

population using the contaminated groundwater was developed by the U.S. 

Nuclear Regulatory Commission (USNRC 1978). Groundwater usage was 

considered to be spatially continuous instead of being from discrete well points. 

The total amount of the released radionuclide ingested by the population 

is 

where 

J

*OO /.OO • /»0 0 

nLJyjQidx*y*. (4.54) 

/ = the ultimate number of curies ingested from the release, 

c = the groundwater concentration (Ci/L), 

Qg = the groundwater withdrawal rate for drinking water 

purposes (m3/d km2). 

If all usage is restricted beyond downgradient distance C from the release 

point, Eq. 4.54 may be integrated in closed form to give 

MoQg US. RdP(\ + y) 

2neRdb

CXp 2DX Dx 

1/21 

(4.55) 

y/X+yiJZ+A-^) 

where 

7 = 

U

2 

4RdDx 

Mb'is the total quantity of the radionuclide discharged to the point source, and 

the other terms are as previously defined: 

If usage of the groundwater is restricted between two downgradient 

distances, J?i and 22, the curies ingested would be defined: 

/ = /(fii)-/(c 2 ), (4.56) 

where /(fij) and 7(fi2) are evaluations of Eq. 4.55 for fi, and fi2 respectively. 

Example 4.5. The use of several of the simpler analytical models in Sects. 

4.5.4 and 4.5.5 will be demonstrated by way of a hypothetical example: 

Leakage into the ground rapidly empties a 1000-ft3 tank containing 4000 

uCi/mL of

 3H, 2000 /*Ci/mL of ^Sr, and 3000 /iCi/mL of 137Cs at a nuclear 

y-
4-42 Radiological Assessment 

site and remains undetected. The site is 50 ft above the mean level and 3000 ft 

upgradient from a river that has representative low flow of 5000 ft3/s and is 

the sink for all surficial groundwater in the area. There are two shallow wells 

located 400 and 2500 ft directly downgradient from the site of the spill. 

Groundwater exists in a homogeneous alluvial sand layer 100 ft thick under 

water table conditions. Dispersivities for the sand have been determined in the 

near field from single-well tracer tests to be 0.5 ft for a.j and 1.0 ft for o^. 

The bulk density pj of the sand is 2.6 g/cm3. Its total porosity n and effective 

porosity ne are 0.4 and 0.25, respectively. The permeability K is 0.02; cm/s. 

Distribution coefficients Kd for the sand have beeh determined to be 0, 2.0, 

and 20.0 mL/g for dilute solutions of

 3H, ^Sr, and 137Cs, respectively. From 

the above information calculate'the following: 

(a) the maximum concentrations of the radioactive components in the river, 

i •. 

(b) the maximum concentrations of the components in the near well, 

(c) the maximum concentrations pf the components in the far well, and 

(d) the total quantity of each radionuclide escaping to the river. 

Solution, (a) If it is assumed that the source is released over a short 

period, Eq. 4.50 for instantaneous releases may be used, to calculate the 

maximum river concentrations of

 3H, ^Sr, and 137Cs. First determine the pore 

velocity U from Eq. 4.8a and the effective porosity ne: 

K 

U = 

AH 

AA 

The gradient 

therefore, 

AH = -5 0 ft 

Ax 3000 ft 

= -0.0167 ; 

TJ - 2 X 10~2cm/s X -0.0167 „ 86,400 s/d , , „ f ' 

U

 dr" — X 30.48 cm/ft

 = 378ft/d

-

The retardation coefficients'for

 3H, ^Sr, and 137Cs can be determined 

from Eq. 4.14: , ' , 

'** " 1 + 14

 x °-° " !

4 • 0.4 

*J~ l + 1 4 X2.0= 14, * 0.4 

,3TCs Rd-> 1 + | 4 x 20 = 131

-

. 0.4 

3H 

"Sr 

Radionuclides in Groundwater 4-43 

The travel times for the three components are calculated by Eq. 4.46: 

3R

 / = A^L = 3000 ft X 1 

"Sr 

U 

3000 ft X 14 

y 

3.78 ft/d

 X 365 d = 2A1 y • 

137, Cs 

3.78 ft/d

 X 3^d " = 304y ' 

3000 ft X 131 „ y _ 

3.78 ft/d

 X 365 d ~ 284 '8

'-

, The half-lives of

 3H, ^Sr, and 137Cs are 12.3 y, 29 y, and 30.1 y, 

respectively. The decay-corrected minimum dilutions in the river are found by 

applying Eq. 4.50: 

2 X 1.0 X 

3H DL = 

5{mix X VT X 1.0 ft X 3000 ft 

3.78 ft/d X 1000 ft X 

86,400 s 

Xexp| ln2 

12.3 y 

= 2.51 X107, 

X2.17y 

90. Sr 

2X14 X 500 °

 ft3 X VirX 1.0 ft X 3000 ft 

DL = 

3.78 ft/d X 1000 ft X 

86,400 s 

Xexp ln 2 

29 y 

6.42X108, 

2 X 131 X 

137, Cs Z>x = 

X 30.4 y 

500 0 ft

 X AT X 1.0 ft X 3000 ft 

3.78 ft/d X 1000 ft X 

86,400 s 

XCXp|30l7 ><284

-

8 y 

. = 2.05X1012. 

The peak concentrations in the river are determined by dividing the tank 

concentrations by the dilution factors: 
4-44 Radiological Assessment 

c(3H) = 4000 /iCi/mL/2.51 XlO7 = 1.59 X 1<F4 ^Ci/mL , 

i • i 

c(90Sr) = 2000 /iCi/mL/6.42X 108 = 3..12X IO"5 nCi/mL , 

c(137Cs) = 3000 MCi/mL/2.05X1012 = 1.46X10~9/iCi/mL . 

(b) Minimum dilution in well (400 ft downgradient). 

First determine whether or not thickness of the aquifer would affect the 

result by calculating the factor <t> from Eq. 4.44: 

* b

1

 = (100 ft)2 

arx 0.5 ft X 400 ft 

= 50. 

Therefore, in this region the release will be relatively unaffected by the 

thickness of the aquifer, and' Eq. 4.48 applies. 

The travel times are estimated using the retardation factors and pore 

velocity calculated above: 

3H 

"Sr 

,37Cs 

400 ft X 1 x 

3.78 ft/d 

y • -

365 d 

400 ft X 14 x y 

3.78 ft/d 365 d 

400 ft X 131 

• 3.78 d 

x y 

X

365d 

0.29 y , 

= 4.06 y, 

= 38 y. 

Applying equation 4.51: 

3H D 0.25 X 1 X (4T X 400 ft)3/2Vl ft X 0.5 ft X 0.5 ft 

L

 ' 2 X 1000 ft3 

Xexp|-^X0.29 y 

22.6, 

»°Sr „ = 0.25 X 14 X (4x X 400 ft)3/Vl ft X 6.5 ft X 0.5 ft" 

L = 2 X 1000 ft3 

XeXP (297 X 4

-

0 6 y 

= 343.6, 

137, Cs _ = 0.25 X J31 X (4r X 400 ft)3/2Vl ft X 0.5 ft X 0.5 ft 

L

 ~ ' 2 x iooo ft3 y

 :

 ~ 

= 6999^9. 

Radionuclides in Groundwater 4-4! 

The peak well concentrations are therefore 177 /*Ci/mL for

 3H, 5. 

MCi/mL for

 90Sr, and 0.43 piCi/mL for l37Cs. 

(c) Well 2500 ft downgradient. Calculate <t> for this region from Eq. 4.44 

<t> = 

(100 ft)2 

0.5 ft X 2000 ft 

= 8.0. 

Therefore^ this well is in the intermediate region, and Eq. 4.48 applies. Tl 

factor F (0) can be read from Fig. 4.7 to be 1.6. Travel times for eac 

component calculated from Eq. 4.46 are 

3H _ 2500 ft X I _JL _ _ g 

3.78 ft/d 365 d '* 

a*, _ 2500 ft X 14 v y _ , , 1 

. " • 3.78 ft/d "365d

 X y' 

•"P., _ 2500 ft X 131 „ y _ ^ . 

' 3.78 ft/d '^365d "''

 y ' 

Applying Eq. 4.48: 

3H 

90o1 X 4ir X 0.25V0.5 ft X 0.5 ft X 2500 ft X 100 ft 

f 1000ft3 X 1.6 

Xexp

 lh 2

 X.1.81y ll2.3 y 'J 

= 271.8 , 

14 X 4B- X 0.25V0.5 ft X 0.5 ft X 2500 ft X 100 ft 

1000 ft3 X 1.6 

Xexpl-!^ X25.2y] 

29y 

= 6275 , 

l37Cs 

i 

_ 131 X4xX0.25V0.5ft X 0.5ft X 2500ft X 100ft 

L

 1000ft3 X 1.6 

Xexp [3o1

f VX237-

4 y 

V: = 7.61 X IO6 . 

The peak well concentrations are therefore 14.7 /*Ci/mL for

 3H, 0.32 pC 

m L for

 90Sr, and 3.9 X 10"4MCi/mL for 137Cs. 

ro 

Ol 
4-46 Radiological Assessment 

(d) Quantity Q of each radionuclide eventually reaching river. 

Equation 4.51 applies to this case because aL « £ (i.e., 1 ft vs 1000 ft). 

Travel times are estimated in part (a) above. The quantity of each radionuclide 

initially in the tank is the concentration multiplied by the volume. Therefore, 

3H Q = 4000/tCi/mL X 1000 ft3

 X 28,300 mL/ft3 

-ln 2 Xexp 12.3v 

X2.17y X10-«Ci//iCi 

"Sr 

= 1.002X10*0, 

Q = 2000 /»Ci/mL X 1000 ft3 X 28,300 mL/ft3 

-In 2 XeXp 29 y 

X30.4y XlO-'Ci/ziCi 

137, Cs 

= 27,370 Ci, '•.•'''.'. ' 

Q = 3bOO,MCi/mL X 1000 ft3 X 28,300 mL/ft3 

i I 

.-In 2 ' Xexp 30.1 y 

X 284.7 y XlO-«Ci/WCi 

120.7 Ci. 

4.6 MODEL VALIDATION 

Hydraulic flow and transport models can be validated (tested) only by 

comparison with field data and, in the 'case of numerical models^ by 

comparison with the analytical solution of a simplified set of equations. In 

general, the model is a set of equations, and validation consists of comparison 

of the solution of these equations with field-measured data. Regardless of 

whether the solution is obtained by analytical or numerical techniques, true 

validation can be done only through comparison with field measurements. 

Agreement of a numerical solution w; h an analytical solution of the same 

equations shows only that the numerical techniques work and that no errors 

exist in the computer code. 

4.6.1 Model Calibration 

Models are calibrated for a specific problem by starting with an initial set 

of parameter estimates (field- or laboratory-measured), running the model for 

the problem, and comparing the results with observed values. If the comparison 

is poor, the parameter estimates are modified, the model is rerun, and results 

Radionuclides in Groundwater 4-4 

are again compared with observed data. This process is continued until tl 

desired level of agreement between observed data and the simulation 

obtained (Mercer and Faust 1980). The modification of boundary conditioi 

and parameters is' subjective and requires a considerable amount of knowledf 

of the region being simulated and experience on the part of the modeler. Tr 

boundary conditions and parameters used in the final simulation must still t 

in agreement with the knowledge and understanding of the geology an 

hydrology of the sitei Through this process the equations of groundwater flo 

in porous media have been well tested and verified. The equations of flo 

through fractured media have been tested to some extent, but the equations < 

transport through fractured media remain largely untested. 

Where enough data on hydraulic heads and variation of head over Ion 

time periods are available, the inverse of the equations for head can be solve 

for the spatial distribution of permeability. The solution of the inverse problei 

(Newman 1973) is useful because it yields a spatial distribution of parametei 

that are consistent with the hydrology of the site under consideration. In th 

process, field-measured parameters are useful for comparison with computei 

, generated parameters to ensure that the generated values are realistic. 

For the transport equations, the inverse problem has not been solve 

because the results are not unique. 

4.6.2 Misuse of Models 

The three most common misuses of models are overkill, inappropriat 

' prediction, and misinterpretation. 

j Overkill is defined as using a more sophisticated model than is appropriat 

for the available data or the level of result desired. The temptation to appl 

the most sophisticated computational tool to a problem is difficult to resist. , 

question that often arises is: when should three-dimensional models be used a 

opposed to twordimensipnal or one-dimensional models? Inclusion of flow in tl 

third dimension, usually vertical, is recommended only in thick aquifers or 

permeability changes, drastically across the thickness of the aquifers. Inclusio 

of the third dimension' requires substantially more data than one- and twc 

dimensonal models. For example, saturated-unsaturated flow through a shalloi 

land burial site is truly a three-dimensional problem. However, the data ar 

seldom available to consider more than one dimension above the water table. 

In many cases, sophisticated models are used too early in analysis of 

problem. One should begin with the simplest model appropriate to the probler 

and program toward the more sophisticated models until the desired level c 

results is achieved. In transport problems the flow modeling should b 

completed and checked against the understanding of site hydrology before 

transport model is applied. 

Misinterpretations usually arise because inappropriate boundary condition 

were selected or the hydrologic history of a site has been misread. Under eithe 

Tl 

ho 

o> 
4-48 Radiological Assessment Radionuclides in Groundwater 4 

of these conditions the simulated data wiil not match the hydrologic history of 

the site. •' . y 

Perhaps the worst misuse of a'model is blind faith in model results. 

Simulated data that contradict hydrologic intuition almost always arise from a 

mistake in some data entry; an error in the computer; code, or application of a 

model to a problem for which it was not designed. The latter case can occur iri 

application of an analytical solution that was, obtained using boundary 

conditions that are different from those to which the solution is being applied 

(Mercer and Faust 1980). , ' 

High-quality field data on contaminant or radionuclide transport in 

groundwater are scarce. The, collection of data necessary for very detailed 

modeling efforts is extremely costly since the aquifer in which the transpprt is 

taking place can be measured only indirectly from wells. Several well-known 

validation efforts are discussed by Roberston (1974b), Codell (1978), Pinder 

(1973), Wilson (1978), Evenson and Dettinger (1980), Anderson (1979), and 

Isherwood (1981). 

4.7 PROBLEMS 

1. Saturated groundwater flow—Determine the seepage velocity for an 

average coarse gravel for a groundwater gradient of 0.002. 

2. Unsaturated flow—A water balance shows that 10 cm/y of water infiltrate 

the ground and recharge the water table. Calculate the average downward 

velocity of the water in the unsaturated zone, which is a fine sand. 

3. Retardation coefficient—Calculate the retardation coefficient for cesium in 

an average sand with a bulk density of 2.8 g/cm3 and Kd of 50 mL/g. 

4. Groundwater concentration—Calculate the concentration as a function of 

time 1500 m directly downgradient of a 1-Ci instantaneous point source in 

an infinitely deep aquifer. The seepage velocity is 1 m/d. The x, y, and z 

dispersivities are 50 m, 20 m, and 1 m, respectively. The effective porosity 

is 0.2. The retardation coefficient is 20. The half-life of the substance is 10 

y5. Dilution in groundwater—For the same conditions in problem 4, consider 

that the 1 Ci was dissolved in 1000 L of water. Calculate the minimum 

dilutionin the well using the equations of Sect. 4.5.4.1. 

6. Dilution in river—For the same conditions as problem 4, calculate the 

minimum dilution in an intercepting river haying an average flow rate of 

10m3/s. 

.'''' . ' • . .. • 

7. Population dose for average usage'—Waste is being discharged to an 

aquifer ingested by downgradient users at a rate of 0.1 m3/(d km2). All 

users are greater than 5000 m and less than 10,000 m downgradient. 

properties of the radionuclide and the aquifer are U = 1 m/d, ax = 

m, ne = 0.2, Rd = 3, b = 100 m, r1/2 = 30 y. Calculate the cu 

ingested for each curie released. 

8. Groundwater flux—Equation 4.51 is an approximation only. The true t 

quantity of flux passing a plane is more accurately determined 

integrating the flux expressed by Eqs. 4.41 or 4.42 from t = 0 to t = 

Graphically integrate the flux from Eqs. 4.41 or 4.42 for a range 

parameters and determine how well Eq, 4.54 agrees. 

ro 

^i 
4-50 Radiological Assessment 

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4-52 Radiological Assessment Radionuclides in Groundwater 

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