ML20214J767

Onsite Disposal of Radioactive Waste.Volume 3:Estimating Potential Groundwater Contamination

ML20214J767
Person / Time
Issue date:	11/30/1986
From:	Ginn T, Dale Goode, Neuder S, Pennifill R NRC OFFICE OF NUCLEAR MATERIAL SAFETY & SAFEGUARDS (NMSS)
To:
References
NUREG-1101, NUREG-1101-V03, NUREG-1101-V3, NUDOCS 8612010449
Download: ML20214J767 (156)

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NUREG-1101 Vol. 3 Onsite Disposal of Radioactive Waste Estimating Potential Groundwater Contamination U.S. Nuclear Regulatory Commission Office of Nuclear Material Safety and Safeguards Daniel J. Goode, Stanley M. Neuder, Roger A. Pennifill, Timothy Ginn ga aece,

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NUREG-1101 Vol. 3 Onsite Disposal of Radioactive Waste Estimating Potential Groundwater Contamination Minuscript Completed: August 1986 DIte Published: November 1986 Daniel J. Goode, Stanley M. Neuder, Roger A. Pennifill, Timothy Ginn Division of Waste Management Office of Nuclear Material Safety and Safeguards U.S. Nuclear Regulatory Commission W shington, DC 20555

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ABSTRACT Volumes 1 and 2 of this report describe the NRC's methodology for assessing the potential public health and environmental impacts associated with onsite dis-posal of very low activity radioactive materials. This volume (Vol. 3) des-cribes a general methodology for predicting potential groundwater contamination from onsite disposal. The methodology includes formulating a conceptual model, representing the conceptual model mathematically, estimating conservative para-meters, and predicting receptor concentrations. Processes which must generally be considered in the methodology include infiltration, leaching of radionuclides from the waste, transport to the saturated zone, transport within the saturated zone, and withdrawal at a receptor location. A case study of shallow burial of iodine-125 illustrates application of the M0CMOD84 version of the U.S. Geological Survey's 2-D solute transport model and a corresponding analytical solution.

The appendices include a description and listing of MOCMOD84, descriptions of several analytical solution techniques, and a procedure for estimating conserva-tive groundwater velocity values.

I NUREG-1101, Vol. 3 iii

CONTENTS Page ABSTRACT.............................................................. iii 1 INTRODUCTION...................................................... 1 2 PROCESSES OF GROUNDWATER CONTAMINATION FROM ONSITE DISPOSAL....... 1 2.1 Infiltration and Leaching.................................... 1

2. 2 Transport Through the Unsaturated Zone....................... 2 2.3 Transport Through the Saturated Zone......................... 2 3 CONCEPTUAL MODELS OF GROUNDWATER CONTAMINATION.................... 3

.4 MEI"000 LOGY FOR ESTIMATING POTENTIAL CONTAMINATION................ 5 4.1 Formulation of an Appropriate Conceptual Mode 1............... 6 4.2 Mathematical Representation..................... ............ 6 4.3 Estimation of Conservative Parameters................... .... 6 4.4 Presentation of Results...... ....................... ....... 8 5 CASE STUDY........................ ......... .... ........... 8 5.1 Conceptual Model.............. .. ...... .. ............ 9

5. 2 Mathematical Representation.............. ... .............. 10 5.3 Parameter Estimation....................... .............. . 10 5.4 M0CMOD84 Simulation.......................................... 11

5. 5 Analytical Solution.......................................... 13 5.6 Concentration in an Onsite Water Well........................ 18 6 UTILITY AND LIMITATIONS OF METH000 LOGY............................. 19 7 REFERENCES........................................................ 20 APPENDICES

A A Numerical Model of Radionuclide Transport in Groundwater, M0CM0084 B October 12, 1983 Update to the USGS Solute Transport Model C Listing of M0CMOD84 Computer Program D Analytical Solutions for Radionuclide Transport in Groundwater E Sensitivity of Radionuclide Transport to Groundwater Velocity

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NUREG-1101, Vol. 3 v

CONTENTS (Continued)

Page FIGURES

~1 Centerline concentrations, MOCMOD84............................... 14 2a Far-field steady-state concentration contours, M0CMOD84 (logio (C))....................................................... 15 2b' Near-field steady-state concentration contours, MOCMOD84.......... 16 3 Concentration 100 ft from source, M0CMOD84........................ 17 TABLES 1 Model parameters, case study...................................... 10 2 Steady-state centerline concentrations , M0CM0084. . . . . . . . . . . . . . . . . . 13-3 Steady-state - centerline concentrations, analytical solution. . . . . . . 18

.4 Effective iodine-123 concentration in onsite water well........... 19 5

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NUREG-1101, Vol. 3 vi

1 . INTRODUCTION Volumes 1 and 2 of this report describe a methodology for assessing the poten-tial public health and environmental impacts associated with onsite disposal of very low activity radioactive materials. Groundwater is a likely transport pathway for radionuclides when such materials are buried. This volume (Vol. 3) discusses a methodology for estimating the potential contamination of ground-water. .This estimate can be used by a licensee, as described in Volume 1, to perform an impact analysis which will be submitted for NRC review.

The methodology for estimating potential groundwater contamination includes for-mulating a conceptual model, representing the conceptual model mathematically, estimating conservative parameters, and predicting receptor concentrations. NRC methodology requires the use of conservative parameters which likely will result in an overprediction of concentrations. When valid site data exist, they are used to determine more realistic parameters, thus refining predictions (see Vol. 1). The cost of obtaining detailed site data for situations that pose potential hazard should be weighed against the cost of alternative disposal.

Herein are described some conceptual models and solution techniques that may be appropriately applied to different sites. An updated version of the modified U.S. Geological Survey (USGS) solute transport computer model is described, as are several analytical solutions for radionuclide transport in groundwater.

One conceptual model is specifically formulated to estimate concentrations for an onsite water well from which some water is drunk, as part of an " inadvertent

-intruder" scenario.

2 PROCESSES OF GROUNDWATER CONTAMINATION FROM ONSITE DISPOSAL Radioactive materials which are disposed of on the ground or under it have a potential to contaminate groundwater in either saturated or unsaturated zones.

The primary mechanisms for contamination are contact of water with the waste from water that infiltrates into the soil or from direct contact with unpack-aged waste; leaching of contaminants from the waste form (animal matter, solids, absorbed liquids); movement of contaminants in water through an unsaturated zone; and movement of contaminants in water through the saturated zone. From the point of view of health risk, the saturated zone is the primary pathway when consid-ering drinking water. However, contamination of the unsaturated zone may be important when one considers intruder scenarios and non-drinking-water pathways as in agriculture. The hydrologic processes related to potential groundwater contamination are summarized in Sections 2.1, 2.2, and 2.3 (cf., Freeze and Cherry, 1979).

2.1 Infiltration and Leaching Waste packages that are disposed of on the land surface (not covered with soil) are directly exposed to rainfall and surface runoff. Waste packages that are buried contact only that water which infiltrates through the overlying soil.

i Infiltration is dependent on precipitation, evapotranspiration (which includes plant uptake), land surface properties, and soil properties. Some of the water NUREG-1101, Vol. 3 1

'Wi that enters the soil will be retained in the soil above the waste and will be evaporated and transpired (through plants). A portion of the infiltrated water will move down to and past the waste packages. In general, infiltration is high in areas with high precipitation, low evapotranspiration, flat or depressed land surface, and sandy soil. Also, infiltration is generally low in areas with low precipitation, high evapotranspiration, steep land surface slope, and silty or clayey soil. In addition, certain inhomogeneities, such as soil fractures, mac-ropores from plants, animals or other stresses, and surface depressions, may increase infiltration dramatically.

Leaching is the removal of contaminants (or other chemicals) from the waste by water moving through the waste. Leaching is affected by the packaging materials, the time of contact between water and the waste, the chemical solubility of the l waste form, and other factors. It is difficult to define general characteristics l of leaching over time and space. It is often assumed that leaching from a buried source is zero until the waste package fails. After such failure, the rate of leaching is initially at a peak value and drops over time. This drop over time can be considered exponential and related to the difference between concentra-tion of contaminants in the waste and in the contacting water. For repeated burial, the rate of leaching may become more constant, especially if burials are frequent relative to characteristic leaching times, and if each burial con-tains a similar inventory.

2.2 Transport Through the Unsaturated Zone if the soil surrounding the buried waste is not saturated with water, contami-nants must move through the unsaturated zone to leave the burial area, or to enter the saturated zone. Characterization of water movement in the unsaturated zone is similar to that of infiltrating water in the soil overlying the waste.

Important differences are that the waste and disturbed soil around the waste may have hydrologic properties different from those of the natural soil, and that the burial area may be a source of increased infiltration. Water movement in the unsaturated zone is controlled by the unsaturated hydraulic conductivity of the soil, the moisture retention characteristics of the soil, the spatial variability of soil properties, sinks and sources, and the location of the water table or saturated zone. The movement of contaminants in the unsaturated zone is affected by dispersion, mobile / immobile water exchange, adsorption, and other physical and chemical processes. This movement is primarily vertical because of high vertical hydraulic gradients. In some cases, especially with layered fine grained materials, horizontal movement can be greater than vertical move--

ment. Transport of contaminants in the unsaturated zone is very sensitive to many of the hydraulic and geochemical properties which are very difficult to predict without extensive site investigations and field measurements. Because of this relative importance and uncertainty, conservative values for screening analyses must be carefully selected.

2.3 Transport Through the Saturated Zone Contaminants which are carried to the water table in percolating water enter the saturated zone. The movement of water and, often, the movement of chemicals in the saturated zone are essentially horizontal. This movement is from areas of groundwater recharge to discharge locations. The latter includes swamps, streams, lakes, springs, and pumping wells. These discharge locations are the receptors for analyses of the health risk posed by drinking water. Vertical

, NUREG-1101, Vol. 3 2

4 processes are important only when strong vertical flow exists because of local flow patterns or deep leakage, or when the contaminants are affected by density or chemical interactions which vary with depth. It is often appropriate to con-sider saturated flow systems at steady state. When considering contaminant transport, the long travel times (perhaps years) through even small aquifers tend to smooth out the effects of varying velocities over daily or even seasonal periods. Special cases of contaminant transport in the saturated zone include immiscible fluids flowing on or through the groundwater without mixing with it, and inhomogeneities such as solution channels which may carry contaminants sig-nificantly faster than average groundwater flow velocities.

3 CONCEPTUAL MODELS OF GROUNDWATER CONTAMINATION i

The conceptual model of a groundwater contamination problem is a qualitative description of the system which contains the important characteristics of the real system and forms the basis for a mathematical or physical simulation of i the system. Primarily, using the conceptual model reduces the problem to man-ageable size, and identifies the relatively important characteristics of the system. A general conceptual model of groundwater contamination includes a contaminant source term, movement of contaminants.in the groundwater system, and withdrawal (and use) of the contaminated water at a receptor. Several ana-lytical solutions for various conceptual models of radionuclide transport in groundwater are given in Appendix D. Generic conceptual models of groundwater contamination are described by many authors, including Codell and others (1982), Aikens and others (1977), and Bear (1979).

The source term incorporates the effects of leaching of contaminants from the i waste form, and movement of contaminated water from the waste location to the groundwater system. In many cases, given the vertical nature of transport in the unsaturated zone, only the saturated zone is considered in the groundwater system, and effects of the unsaturated zone are included in the source term.

Spatially, the source may be described as a point, line, area, or volume of material. The dimensions of the source should be consistent with the dimen-sions of the groundwater system. For example, if vertical processes are ig-nored, the source should be a horizontal area rather than a volume. For a given inventory, the smaller a source is considered, the higher the calculated peak concentrations are. On the other hand, smaller sources also present smaller areas of contamination. The release of contaminants to the groundwater system over time must also be defined. Three common conceptualizations of the time variation of the source are: instantaneous release after burial or fail-

ure; a continuously decreasing release over time; or, a constant rate of re-lease until the entire available inventory is gone from the waste form. As above, instantaneous releases have the highest peak concentrations, but con-stant rate releases have lower concentrations that last a longer time. The temporal nature of the source should also be consistent with the time frane of interest. For example, if annual average water concentrations are used as the screening criterion, the most conservative conceptual model contains the tem-poral description that yields highest annual averages, not necessarily highest peak concentrations.

The conceptual model must include radionuclide transport through the saturated and unsaturated zones of the groundwater system. As mentioned above, a common approach for the unsaturated zone is to include it as a time-delay release in the source term. In this fashion, the radionuclides would be assumed to enter NUREG-1101, Vol. 3 3

4 the saturated groundwater system at some time after burial, which includes both retention in the waste package before failure, and movement through the unsat-urated zone. This conceptual model assumes that transport in the unsaturated zone is essentially. vertical and does not spread radionuclides horizontally.

Alternatively, the unsaturated zone may be considered as a part of the saturated zone with, perhaps, a different travel time (Aikens et al., 1977). In addition, depending on the nature of soil at the site, it may be appropriate to incorpor-ate the adsorption of radionuclides on the soil in the unsaturated zone. Rea-sonable estimates of the impact of these processes may be difficult to make in the absence of site data. A conservative model, or one that is likely to over-predict the rate of radionuclide movement, may ignore the unsaturated zone and assume that radionuclides released from the waste package enter immediately into the saturated zone.

Many processes affect the movement of radionuclides in groundwater. Advection is the process of radionuclides being carried in the flowing groundwater. In the absence of other processes, radionuclides are expected to move at the same rate as the water moves. However, laboratory and field studies show that non-reactive tracers in groundwater do not i11 move at the average groundwater veloc-ity. Because of the tortuous flowpath and the spatial variability of porous media flow, a concentration front in groundwater disperses, or is smeared out.

Thus, some of the radionuclides move faster than the average groundwater veloc-ity, and some move slower. This process is termed dispersion and includes mix-ing caused by several different system properties (cf., Freeze and Cherry, 1979; Bear, 1979). In addition, many radionuclides react with the soil by adsorbing on the soil surface. Since the adsorption process is typically very fast, com-pared to groundwater flow rates, it is often assumed that the amount of radio-nuclide adsorbed on the soil surface is in equilibrium with the amount of radio-nuclide in the water, and that this equilibrium occurs instantly. This effect causes the radionuclides to appear to move more slowly than the average ground-water moves, hence the term retardation. The retardation factor is the ratio of the average groundwater velocity to the radionuclide transport velocity, and can be much greater than 1. Some radionuclides may react with other chemical species (e.g., EDTA) to produce chemical complexes with different mobility than the radionuclide alone (Robertson et al., 1984). For radionuclides with rela-tively short half-lives (most radionuclides considered for onsite disposal),

significant decay can occur during the relatively slow transport in groundwater, and should be considered in the transport estimation. Available mathematical models of radionuclide transport in groundwater (cf., Appendix D) include ground-water velocity (advection), dispersion, retardation, and decay. For some cases, appropriate conceptual models allow some of these processes to be removed from the mathematical model with no significant loss of accuracy. In addition, this minor increase in error may be relatively unimportant when compared with errors in soil hydraulic and geochemical properties.

The general equation for radionuclide transport in groundwater is three dimen-sional. All three dimensions must be included if flow or transport occurs in all directions. For many real situations, however, it is sufficient to consider only one or two dimensions. For example, many shallow groundwater systems are very thin, compared to the distance from the source to the receptor. In this case, the radionuclides may be considered completely mixed over the vertical and at any location the concentration does not vary with depth. Even if there is no vertical flow and the source is at the top of the saturated zone, disper-sion causes the radionuclides to spread into the entire saturated zone. The NUREG-1101, Vol. 3 4

distance at which the mixing is essentially complete can be estimated from the dispersion parameters and the characteristics of the. source (Bear, 1979). With-out detailed information about the site, it is usually appropriate to assume that the flow system is uniform; that is, that velocities are everywhere equal.

This implies that there is no local recharge. In reality, flow is not uniform, but responds to local variations in soil properties, recharge, and geometry of the aquifer and its boundaries (e.g., streams). For complex systems with non-uniform flow, it may be necessary to use numerical techniques to solve the gov-

, erning transport equations (cf., Appendix A).

The conceptual model must also include the temporal nature of the radionuclide transport problem. For groundwater transport, the flow system may not change significantly over time. In addition, the slow movement of radionuclides over several months or years will smooth the effects of short-term fluctuations in the hydrologic system. Thus, it is often appropriate to assume that the flow system.is at steady state. For some situations, it may also be sufficient to simulate the steady-state behavior of the radionuclides and ignore the rela-tively short time period during which a plume develops. This approach assumes that the mass of radionuclides being continuously added at the source is exactly

-balanced by the amount discharged and by the amount that decays in the ground-water system over a given time period. Transient development of the plume must be considered when the source term varies in time, or the source duration is not sufficient for a steady-state plume to develop.

The third component in the conceptual model of radionuclide transport in ground-water is the receptor, and its impact on that transport. The receptor is the location at which the impact of radionuclide concentrations in groundwater is evaluated. This impact may be simply the environmental concentration resulting in a degradation of the quality of groundwater, or it may be an estimated human dose resulting from ingestion of food grown at the receptor location. For drink-ing water impacts, the receptor may be a pumping well or a stream that receives groundwater discharge and that is also a source of drinking water. Some recep-tors, especially pumping wells, may have a significant effect on the groundwater flow system and radionuclide transport. Depending on the characteristics of the system, this effect may be an increase or reduction in calculated peak or average concentrations. In some cases, these effects may be approximated with-out considering modification to the general conceptual model (Aikens et al.,

1977). Streams are conveniently included in the conceptual model of the ground-water system. If the withdrawal rate of the receptor is small compared with the groundwater flow rates, it may be ignored in the flow simulation and the resulting concentration at that location may be used as the receptor concentra-tion for impacts analysis.

4 METHODOLOGY FOR ESTIMATING POTENTIAL CONTAMINATION j

The methodology for estimating potential groundwater contamination from onsite disposal includes formulation of a model of radionuclide transport and is out-lined as:

formulation of an appropriate conceptual model of the flow and radionuclide transport system j -

mathematical representation of the conceptual model i

NUREG-1101, Vol. 3 5 i

- estimation of conservative parameters from available site data and generic information simulation and analysis of results There is feedback between all the steps of this methodology and the entire process may be performed several times as new information becomes available.

4.1 Formulation of an Appropriate Conceptual Model An appropriate conceptual model is selected on the basis of available informa-tion. The conceptual model must specify the nature of the source term

- the characteristics of flow and transport through the unsaturated zone the dimensionality and geometry of the saturated zone the location of groundwater recharge and discharge areas the general directions and rates of groundwater flow the importance of advection, dispersion, adsorption, and retardation

- the nature of the receptor and its impact on flow and transport

- the spatial variability of flow and other characteristics

- the transience of the source, flow' system, and receptor 4.2 Mathematical Representation The radionuclide concentration at a receptor is estimated by solving the mathe-matical representation of the conceptual model with the estimated parameters.

For many conceptual models, the solution is analytical and may be directly cal-culated (cf., Appendix D). For others, particularly with complex geometries and non-uniform parameters, a numerical solution may be required (e.g., Appen-dix A). As necessary, numerous solutions can be generated with ranges of param-eters that may realistically represent the field situation. In all cases, it is advisable to perform several calculations to verify that the computed concen-trations vary as expected with changes in the input data. For numerical models, the model should be tested by comparison with available analytical solutions.

For applications for which there is no analytical solution, the user should carefully note mass balance calculations, if performed, and the general nature of the computed concentration distributions. Graphics are often useful for checking model consistency of both analytical and numerical solutions.

4.3 Estimation of Conservative Parameters In the absence of site-specific information, conservative components of the con- ,

ceptual model are chosen on the basis of the impact measure (peaks or averages) and the interrelation between system components. In general, a conservative conceptual model for peak concentrations includes an instantaneous release to the saturated zone, no geochemical interactions between the radionuclidcs and the soil or rock, and one-dimensional flow and transport with no dispersion.  ;

For annual concentrations (or doses), a constant-release source, with all radio- I nuclides released in one year, gererally yields conservative estimates. l Parameters for the model are estimated frcm available site information and gen-eric information on similar media. Site information required from the licensee is described in Volume 1. Since most onsite disposals will take place at NUREG-1101, Vol. 3 6

facilities which are not primarily used for waste disposal, it may be that little site information is available, other than general descriptions and geometric characteristics. Without site data, the model parameters must be conservatively estimated on the basis of properties of similar media in similar geologic set-tings. Appendix B in NUREG-1054 (Codell, 1984) contains examples of generic flow and transport model parameters. If use of conservative or " worst case" values results in predictions of significant contamination, it may be necessary to obtain site data that will allow more realistic predictions. Parameters that may be required, on the basis of the conceptual model chosen, include locations of the source and potential receptors geometry and duration of the source (where and how often is the waste buried) concentrations and amounts of radioactivity in waste waste package lifetime thickness, moisture content, and hydraulic conductivity of the unsaturated zone thickness, porosity, and hydraulic conductivity of the saturated zone recharge rate hydraulic gradient radionuclide travel time longitudinal and transverse dispersion coefficients adsorption and retardation factors decay rate withdrawal rate at receptor and at other points other model-specific parameters In addition to conservative values, it may be appropriate to estimate a range of parameters that brackets expected behavior of the real system. When estimat-ing parameters, one should consider that the system parameters are not indepen-dent, but may be directly related to each other, and assumed values should be consistent.

Velocity can be estimated through water balance considerations in the absence of site-specific data. For conservation of mass over a long time period, the water that recharges a groundwater system must flow through and out of the sys-tem. A long-term average groundwater discharge can be estimated as the average recharge rate times the area of recharge. For example, if the average annual recharge is 10 cm, and the point of interest is 500 m from a groundwater divide, then the average discharge per unit width at that point would be 10 cm/yr x 500 m NUREG-1101, Vol. 3 7

or 50 m3 /yr/m. The velocity is this discharge divided by the saturated thickness and the porosity. Assuming a saturated thickness of 10 m and a porosity of 0.3, the velocity for this example is (50 m3/yr/m)/10 m/0.3 = 16.7 m/yr (54.7 ft/yr)

A similar approach can be used to estimate vertical velocity in the unsaturated zone as V = R/M, where R[L/T] is the average recharge rate and M[L3/L3] is the average moisture content in the unsaturated zone.

Although the sensitivity of calculated concentrations to model parameters is site specific, the general sensitivity of these concentrations to some param-eters can be described. If aquifer discharge is fixed and velocity is changed, then velocity changes correspond to porosity changes. In this case, increasing velocity. increases calculated receptor radionuclide concentrations. This is because the radionuclide decays less in the reduced time it takes to reach the receptor. If, however, increases in velocity are considered to imply an in-crease in discharge, this effect is offset by dilution which reduces coricentra-tions. Appendix E discusses this case, and describes techniques for estimating a critical velocity which will result in maximum peak concentrations. The re-tardation factor accounts for an apparent reduction in the velocity of radio-nuclide movement owing to adsorption and other chemical processes. Since a i reduction in retardation increases apparent velocity and does not affect dis-charge, that change results in increased radionuclide concentrations at the receptor. Retardation is a function of the adsorption coefficients, bulk den-sity, and porosity. In addition, it may be a function of concentration.

4.4 Presentation of Results l

The licensee's presentation of final results should include a description of the conceptual model, the mathematical representation, all parameters, and calculated concentrations. All generic data should be referenced and all information used to estimate site-specific parameters should be presented and discussed. Graphic plots of concentration versus time or plots of concentration on a site map may be useful, depending on the temporal and spatial nature of the conceptual model.

5 CASE STUDY t

l This case study shows how the general procedures described above are applied in

! evaluating an onsite disposal operation. This case does not prescribe informa-

> tion to be submitted, but it does illustrate some of the more important proce-4 dures and results for a particular situation.

. The waste disposal operation and general site characteristics were described as follows in the licensee's application for license modification i l The licensee operates a biological research facility in the northeast i I

U.S. Animal carcasses and trash contaminated with trace quantities of short-lived radionuclides are buried four times a year. None of the radionuclides are in bulk liquid form. Iodine-125, the radionuclide present in the largest quantity, has one of the slowest decay rates of the inventory. Only the analysis of iodine concentrations is pre-sented below. Less than 750 ft3 of waste is buried each year and any

NUREG-1101, Vol. 3 8

one burial will contain less than 0.05 Ci of iodine. For each l burial, a shallow trench approximately 10 ft x 15 ft in cross section is dug with a backhoe and the waste, in 4-mil-thick plastic bags inside of fiberboard drums, is placed in the trench and covered with at least 6 ft of fill. The water table is approximately 10 ft below the bottom of the buried waste. The licensee estimates that. i groundwater flows through the rocky soil at a velocity less than 1 ft/ day. The groundwater discharges to a swamp about 2/3 mile from the burial trenches. The licensee included a site map indicating the location of the burial trenches and other site features.

5.1 Conceptual Model

.A conceptual model, based on the general characteristics of the site and the region, is chosen. Although the waste package described probably isolates waste from the environment for a significant time, very little information about the

' subsurface integrity of the 4 mil-thick plastic inside the fiberboard is avail-able. For a conservative analysis, it is considered appropriate to ignore re-tention by the waste package. Since annual doses are the impact of interest, the source is considered to be continuous release of all radionuclides buried 4

during a year within a year. The daily rate of release is thus the annual bur-ial quantitity divided by 365 days. This source rate is probably much higher than the actual rate. The source is considered a point, which will maximize centerline concentrations while reducing the width of the contaminated zone close to the source.

In the northeast, the water table is typically shallow. In addition, precipi-tation is high and most recharge discharges to local surface water flow systems, i including swamps. In this climate, radionuclides can potentially move through l

the unsaturated zone very rapidly. For this reason, it is considered appropri-ate to ignore retention in the unsaturated zone. Within the saturated zone, vertical processes are ignored because there is a lack of site-specific informa-4 tion. Fully penetrating wells and dispersion, for many cases, will effectively 4

mix radionuclides over the vertical. In order to ensure a conservative estimate of the rate of movement through the saturated zone, the aquifer is considered to be relatively thin and to have a high discharge rate. These data are de-I scribed below.

Ignoring the vertical dimension, transport through the saturated zone is con-1 ceptualized as two-dimensional advection and dispersion with linear adsorption and radioactive decay. This conceptual model treats the aquifer as a porous j

medium; this treatment may not be appropriate for bedrock aquifers with signi-j ficant fracturing. Initially, the problem is considered a transient one. How- I ever, results demonstrate that for the short-lived radionuclides of interest, a J.

steady state is reached rapidly. A linear equilibrium adsorption isotiierm is assumed because it has been shown to be applicable to many geochemical situa-tions and is accurate at low concentrations, and because most generic data are

, presented in a form assuming that isotherm. Although there are no actual re-ceptors between the burial area and the swamp, the receptor is considered to be a well near the property line of the facility. This well is assumed to have no effect on the uniform flow field. In a system that receives recharge along j its flowpath, discharge through the aquifer must increase toward a discharge i

i NUREG-1101, Vol. 3 9 i

point. As above, this was not included because site-specific information was lacking. Ignoring this effect and using a high uniform velocity will likely result in higher concentrations since velocities are high near the source, as well as near the discharge point.

5.2 Mathematical Representation The mathematical representation of this conceptual model is a standard one in groundwater hydrology (cf., Bear, 1979; Freeze and Cherry, 1979; Fried, 1975).

The governing equation is a

-V -R d (1)

R d * "xVh+aV y AC with initial condition C (x, y, t = 0) = 0 and boundary condition C (x = 1 m, y = =)=0 and a continuous constant injection rate f' at x=y=0 The terms in equation 1 are definea in Appendix D and Table 1. This equation includes two-dimensional dispersion in a uniform flow system with flow in the x direction. It also includes linear equilibrium adsorption and radioactive decay. The parameters which must be determined for this model are source rate, groundwater velocity, porosity, thickness, longitudinal dispersivity, transverse dispersivity, retardation factor, and the radioiodine decay rate. The values assumed for this analysis are shown in Table 1.

Table 1 Model parameters, case study Parameter Value f' Iodine-125 source rate (constant) 0.2 Ci/yr A Decay rate 4.216 yr 1 V Groundwater velocity (uniform) 365 ft/yr n Porosity 0.1 b Aquifer saturated thickness 10 ft a*

Longitudinal dispersivity 20 ft Transverse dispersivity 4 ft Distribution coefficient 0.1 ml/g a{

K p Soil solids density 2.4 g/ml R

s Retardation factor 3.16 NUREG-1101, Vol. 3 10

5.3 Parameter Estimation The source rate is estimated as the maximum amount of iodine per burial times 4 burials per year (0.05 Ci x 4/yr = 0.2 Ci/yr). Since these amounts are maximums, this rate is higher than the actual annual average rate. However, a short-term release could decur at a higher rate. No additional processes are assumed to affect the release of iodine to the saturated groundwater system.

The groundwater velocity is estimated as 1 ft/ day, or 365 ft/yr. This corre-sponds to the applicant's information and is a conservatively high velocity.

Using the estimation technique described above, a groundwater velocity of 1 ft/ day corresponds to 12 in./yr recharge and a point 365 ft from the ground-water divide. The critical velocity for iodine-125 in this system is estimated l to be about 9 ft/ day (cf., Appendix E) because of the short half-life. Assum- l ing a porosity of 0.1 results in a relativly small system discharge which, in l turn, results in lower dilution and higher concentrations. Porosity also affects the calculation of the retardation factor. The aquifer thickness is assumed to be 10 ft which is relatively thin for surficial deposits in the northeast. This assumption, although somewhat arbitrary, reduces dilution and increases concen-trations. Since the width of the contaminated zone is of little interest here ,

compared to peak or centerline concentrations, relatively low dispersivities of l 20 ft (longitudinal, x) and 4 ft (transverse, y) are assumed. Higher values would flatten the plume and result in lower peak concentrations. Isherwood

~

(1981) presents longitudinal dispersivities determined by field scale modeling ranging from 10 ft to more than 600 ft.

The retardation factor is computed as (1 n)p Rd=1+ n K

d (2) which assumes linear equilibrium adsorption. The terms in equation 2 are de-scribed in Table 1. Staley and others (1979) report a conservative distribution coefficient Kd f r iodine of 0.1 ml/g. Using this value, a solid's density of the soil of 2.4 g/ml, and the porosity above, the retardation factor is calcu-lated as 3.16. The iodine has an apparent velocity about three times less than the groundwater velocity, or about 1/3 ft/ day.

Two primary techniques are available for generating solutions to governing equa-tions for radionuclide transport in groundwater: analytical solutions and num-erical models. Both techniques are used below to simulate potential contamina-tion for the case study, illustrating some of the differences between the two methods.

5.4 M0CM0084 Simulation The MOCMOD84 computer model for radionuclide transport in groundwater (cf. ,

Appendix A; Konikow and Bredehoeft, 1978; Tracy, 1982) can be used to simulate transport for the case study. Since the flow field is assumed to be uniform, and only one source is considered, the centerline of the plume is taken as a line of symmetry. Thus only one-half of the spatial domain is modeled. This domain is bounded by fixed head boundaries upgradient and downgradient of the l

NUREG-1101, Vol. 3 11

i plume, and by no-flow boundaries at the centerline and a reasonable distance from the source. The distances between the source and these boundaries are adjusted in such a way that concentrations are small near all boundaries, ex-cept at the centerline (cf., Appendix A). A regular grid with ax and ay of 100 ft was initially specified. Results presented below are generated with a grid spacing of 25 ft. This discretization produces satisfactory results and maximizes the number of cells inside the plume within the computer storage limits.

Many of the physical properties described above are directly specified in the model input dataset (Appendix C). In the model, the groundwater velocity is i determined by solving the steady-state equation for two-dimensional groundwater flow with specified boundary conditions. Thus the velocity cannot be directly specified. Instead, boundary conditions and hydraulic conductivity are speci-fied so that the calculated velocity (and discharge) agrees with the desired ~

value. For convenience, flow is specified in the y direction only (in the out-put in Appendix C, flow is down the page). A head gradient of 0.01 is imposed by fixed-head boundary conditions upgradient and downgradient of the source.

In M0CMOD84, fixed head boundary conditions are simulated by leaky cells with very high leakage coefficients so that the aquifer head is essentially equal to the head in the leakage source / sink. The model grid is 42 cells long (in the direction of flow, y) and 10 cells wide. On all the boundaries, the model re-quires a row of no-flow cells; thus, the flow domain is 40 cells x 8 cells.

The leakage source / sink heads in rows 2 and 41 are specified as 10.0 and 0.25 ft, respectively, resulting in the desired head gradient of 0.01. As discussed above, the saturated aquifer thickness is 10 ft and the porosity is 0.1. For a groundwater velocity of 1 ft/ day and a porosity of 0.1, the discharge flux (Darcy velocity) is 0.1 ft/ day. The hydraulic conductivity is specified as 10 ft/ day so that a head gradient of 0.01 results in this flux. For the model, transmissivity (hydraulic conductivity integrated over saturated thickness) is specified as 100 ft/ day (1.15741E-3 ft/sec).

The radioiodine source is simulated as an injection well in one cell (25 ft x 25 ft). The rate of water injection is specified as SE-7 ft3/sec, small enough so that it does not significantly affect the flow field. To balance this volume, a withdrawal well with the same rate is located directly downgradient from the source at the last active cell. The mass of radio-nuclides removed by this well is essentially zero since the concentration is negligible at this point. The concentration in the injected water is specified so that the output concentrations of interest are greater than 0.1 and less than 10,000. This is because of the format of the computer model printout. Since i

the problem is linear (including the adsorption isotherm), the output concentra-tions can be scaled by the ratio of the actual source rate to the modeled source rate. For example, if the model source rate is 1E9 Ci/yr and the actual source rate is 0.2 Ci/yr, the output concentrations are scaled by 0.2/1E9 = 2E-10.

lhis scaling may not be appropriate if a nonlinear isotherm is used. Since only half of the flow field is simulated, the model input rate (5E8 Ci/yr) is half the rate for the entire system.

The simulation results indicate the transport of radiciodine in time and space.

Steady state is reached within 2.5 years at all centerline points within 500 ft of the source (Figure 1). This, of course, assumes that the source term is 1

NUREG-1101, Vol. 3 12 l

i

{ _ _ _ _ _ _ _ _ _ _ _ _ . _ _

constant for that period. For the source rate of 0.2 Ci/yr, the peak concentra-tions for four points downgradient from the source are shown in Table 2. Beyond about 500 ft, relative concentrations are very small because the decay of the radiciodine (Figure 2).

Table 2 Steady-state centerline concentrations, M0CMOD84*

Distance downgradient Approximate concentration (ft) (Ci/ft3) 50 1.1E-6 100 3.0E-7 250 s 6.7E-9 500 -1.4E-11

• Example: Iodine-125; source rate = 0.2 Ci/yr Numerical " noise" in the model output is caused by the moving particle tech-nique employed by MOCM0084. This is illustrated by a linear scale plot of relative concentration 100 ft from the source,(Figure 3). The M0CMOD84 solu-tion agrees in general with the nature of the analytical solution (discussed below). However, care must be taken in selecting model output values for deci-sionmaking. As shown in Figure 3, the concentration varies by up to 20% after steady state has been reached. Simulation of 2.5 years required 39 particle moves and resulted in a chemical mass balance error of 4.4%. s 5.5 AnalyticalSolutio$

Fortunately, an analytical solution is available for the governing equation and boundary conditions for this example case. This solution is described in Appendi.x D (Wilson and Miller, 1978, 1979). As shown~above with M0CMOD84, the system rapidly reaches steady state. For steady state and sufficient distances .

downgradient, Wilson and Miller (1978,1979) present an approximate solution g(x, y) , f' exp(x/B) exp -r/B) when r/B > 1 (3)

(8n/8)b nbV(a x a ) (r)

The terms in equation 3 are described in Appendix D (p. D-13).

For the case study, the physical parameters of the system are directly specified, including groundwater velocity. This analytical solution assumes that the flow field is ' infinite in all directions. The centerline concentrations at four dis-tances downgradient from the source are shown in Table 3. These values agree with the values in Table 2 from M0CMOD84.

Some of the differences between the M0CM0084 and the analytical solution can be attributed to the approximations that each makes. The source in the M0CMOD84 simulation is an area of 25 f t x 25 ft, whereas the source for the analytical l

l NUREG-1101, Vol. 3 13

o O

8:

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o

~

-cccc e s=======c,ea==e;  ;

8, z-o -

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o o 50 FT ta n 100 FT p! o 250 FT a-  : a 500 FT J _

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M-ch ^ a a" i.

~

a. -

o I I I B

0.0 0.5 1.0 1.5 2.0 2.5 YERRS l

Figure 1 Centerline concentrations, MOCM0084 NUREG-1101, Vol. 3 14

STEROY STATE CONCENTRATIONS (LOG 101 o.

to 5 ,

,/ -2 ~, '.

/ ,

o l / ', '. '

S- / '. i 1 j

- i i u  ; ~, '

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0.0 100.0

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\

400.0 500.0 600.0 I.

700.0 I.800.0 i 900.0 .

LONGITUDINRL DISTRNCE NOTE: Not drawn to scale.

I

(

Figure 2a Far-field steady-state concentration contours, MOCM0084 (logio (C))

NUREG-1101, Vol. 3 15

STEADY STATE CONCENTRATIONS

!d R-W 1" -

h.

~~

r E 10 -

U j$* 100 c:3 1000 g _-,' . ' ' .'.

ICO.0 lb.0 250.0 2d.0 3d0.0 35'.0 0 400.0 LONGITUDINAL DISTANCE NOTE: Solid lines above 4000 drawn at intervals of 4000.

l i

Figure 2b Near-fieldsteady-stateconcentrationcontours,M0CMdD84 l

NUREG-1101, Vol. 3 16

8 8

z o 8_

- ui e- ,

a-e E-z l- w Oc za_

l oo l u-u s

ac a 100 FT d Ol~

m L

o :;-

i i i i l 0.0 0.5 1.0 1.5 2.0 2.5 YERRS Figure 3 Concentration 100 ft from source, M0CMOD84 NUREG-1101, Vol. 3 17

Table 3 Steady-state centerline concentra-tions, analytical solution

• L Distance downgradient Concentration (ft) (Ci/fts) 50 2.3E-6 100 4.7E-7 250 7.56E-9 500 1.2E-11

• Example: Iodine-125; source rate = 0.2 Ci/yr solution is a point. Thus, concentrations on the centerline for the analytical solution should be higher near the source. The observed differences betueen solutions decrease with distance from the source (cf., Tables 2 and 3). Un addition, the analytical solution used here is approximate. Its accuracy

• increases with distance from the source (Wilson and Miller, 1978). .

I The analytical solution has several advantages over the numerical model sofu-tion. The analytical solution is directly reproducible and can be computee without extensive computer resources or experience. It also has no numerical noise nor any execution parameters that must be specified by the user. Thel limitations of the analytical solution are that it cannot incorporate non-uniform aquifer properties or arbitrary boundary and source conditions. This limitation includes the fact that aquifers with variable properties and geometry cannot be simulated. For complex geometries and inhomogeneous systems, use of numerical solutions is often more attractive, if not mandatory.

The results of both solution techniques are steady-state concentrations of iodine downgradient from the waste disposal location (source). These concen-trations can be used to perform a conservative risk assessment for various ex-posure scenarios such as agriculture and drinking water withdrawal.

5.6 Concentration in an Onsite Water Well The methodology is also applied to the scenario of an inadvertent intruder who obtains drinking water from an onsite well. The conceptual model considers immediate release of the entire radionuclide inventory (accounting for decay) at some time after the last burial. This mass is completely dispersed in 91,250 liters of groundwater. This is the annual volume of water from a domes-tic well used by one person for all purposes (250 liters / day) (Miller, 1980).

The radioactivity concentration is the total radioactivity divided by 91,250 liters. A portion of this water is drinking water, and the concentration in that water drops over time.

The effective concentration in drinking water for an annual dose is obtained by integrating the concentration over a 1 year period. The mathematical represen-tation of this model is NUREG-1101, Vol. 3 18

M C, = (1 yr)(91,250 liters) eXP(-At)dt g

M l

• (1 yr)(91,250 liters) [1 - exp(-A)] (4)

Where M,[Ci] is the total radioactivity at the time of release of the site, A[yr 1] is the decay rate, and C,[Ci/ liter] is the effective concentration.

For iodine-125, the inventory immediately after burial is the radioactivity for that burial plus the undecayed portion of prior burials M,= 0.05 + 0.1174 + 0.0061 + 0.0021 ...

3 0.076 Ci The effective concentration C, for different site holding periods (before release after last burial) is shown in Table 4.

Table 4 Effective iodine-123 concentration in onsite water well Effective concentration, Holding period (yr) C3 (pCi/ liter) 0 19500 1 2900 2 42.9 3 0.63 For short-lived radionuclides (such as iodine-125), this methodology may deter-mine minimum holding periods before sites can be released for unrestricted use.

For longer-lived radionuclides (carbon-14), this methodology may result in a maximum burial inventory for a given area. Other processes, such as leach re-sistance and pumping efficiency, could be included on the basis of site-specific information.

6 UTILITY AND LIMITATIONS OF METHODOLOGY Several limitations should be considered during application of this methodology.

The conceptual model of potential contamination is an approximation of reality and may not incorporate important phenomena. The physical parameters used in the screening model may not be conservative for the particular site. These un-certainties encourage using conservative or " worst case" parameters and models, where possible. Site data allow the use of more realistic parameters, but it should be noted that single point measurements may not incorporate the overall

, behavior of the transport system.

NUREG-1101, Vol. 3 19

Appropriate selection of models and parameters can provide reasonable assurance that predicted concentrations will not be exceeded in reality. When the use of conservative models and parameters results in significant potential risk, fur-ther site data may be warranted. The effort expended in collecting additional data should be weighed against the cost of alternative disposal, at a commercial low-level waste disposal facility, for example.

7 REFERENCES Aikens, A. E..Jr., R. E. Berlin, J. Clacy, and 0. I. Oztunali; " Generic Method-

-ology for Assessment of Radiation Doses From Groundwater Migration of Radio-nuclides in LWR Wastes in Shallow Land Burial Trenches," Atomic Industrial Forum, AIF/NESP-013, 1977.

Bear, J., Hydraulics of Groundwater, McGraw-Hill, New York, 1979.

Codell, R., " Simplified Analysis for Liquid Pathway Studies," U.S. Nuclear Regulatory Commission, NUREG-1054, 1984.

Codell, R. B., K. T. Key, and G. Whelan, "A Collection of Mathematical Models for Dispersion in Surface Water and Groundwater," U.S. Nuclear Regulatory Com-mission, NUREG-0868, 1982.

Freeze, R. A., and J. A. Cherry, Groundwater, Prentice-Hall, Englewood Cliffs, New Jersey, 1979.

Fried, J. J., Groundwater Pollution, Elsevier, Amsterdam, The Netherlands, 1975.

Isherwood, D., "Geosciences Data Base Handbook for Modeling a Nuclear Waste Repository," U.S. Nuclear Regulatory Commission, NUREG/CR-0912, 1981.

Konikow, L. F., and J. D. Bredehoeft, " Computer Model of Two-dimensional Solute Transport and Dispersion in Ground-water," U.S. Geological Survey Techniques of Water Resources Investigations," Book 7, Chapter C2, 1978.

Miller, D. W. (ed), Waste Disposal Effects on Ground Water, Premier Press, Berkeley, California, 1980.

Robertson, D. E., A. P. Toste, K. H. Abel, and R. L. Brodzinski, "Radionuclide Migration in Groundwater," U.S. Nuclear Regulatory Commission, NUREG/CR-3554, 1984.

clide Migration From Staley, G. B. , G. P. Turi, and A Generic D. L. Screiber, Overview," "Radion(f in Management o E6 Leyl Radioactive Low-Level Waste:

Waste, Pergamon Press, New York, 1979.

Tracy, J. V., " Users Guide and Documentation for Adsorption and Decay Modifica-tions to the USGS Solute Transport Model," U.S. Nuclear Regulatory Commission, NUREG/CR-2502, 1982.

Wilson, J. L., and P. J. Miller, Two-dimensional Plume in Uniform Groundwater Flow," J. Hydraulics Division, ASCE, 104(HY4):503-514, 1978.

NUREG-1101, Vol. 3 20

j.

Wilson, J. L., and P. J. Miller, "Two-dimensional Plume in Uniform Groundwater Flow, Closure,"'J. Hydraulics Division, ASCE, 105(HY12): 1567-1570, 1979.

i I

i 3

i

'l NUREG-1101, Vol. 3 21

% ,.a&6., h _,22 - - - 44,. 5,A,, _ , * - e d-)%d. ,_4 4 Om , p .--A,,1 4,.mg .m. 4 A= J S-- a.

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<- l f

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f APPENDIX A A NUMERICAL MODEL OF RADIONUCLIDE TRANSPORT IN GROUNDWATER, M0CM0084 INTRODUCTION The USGS (U.S. Geological Survey) two-dimensional solute transport and disper-sion computer model (Konikow and Bredehoeft,1978) has been modified to account for retardation due to sorption and radioactive decay (Tracy, 1982). Steady-state or transient groundwater velocity is calculated from a finite difference solution of two-dimensional horizontal groundwater flow. Applying the method of characteristics, solute transport is split into two components for computa-tional purposes: (1) advection - transport in the flowing groundwater, and (2) dispersion - spreading due to diffusion, hydrodynamic dispersion, and macro-scopic dispersion. Dispersion is evaluated using standard finite difference techniques; advection is simulated with a moving point technique. Advected solute " particles" move along groundwater streamlines or characteristic curves.

The computer model discussed here allows the simulation and prediction of radio-nuclide concentrations in sorbing aquifers with complex geometry, non-homogeneous properties, and non-uniform boundary conditions and source terms.

In addition to Tracy's transport modifications (1982), the computer program has been modified by the original authors in a series of updates. The most recent update, October 12, 1983, is attached as Appendix B and those changes have been incorporated in the program listing in Appendix C. Except for these updates, this version, M0CMOD84, is essentially the same as M0CMOD described by Tracy (1982).

This appendix is a summary of the original M0C user's manual by Konikow and Bredehoeft (1978) and the user's manual for the modified version, M0CMOD, by Tracy (1982). The user must rely on these documents for detailed discussion and input data descriptions.

CALCULATION OF GROUNDWATER VELOCITY Groundwater flow is determined as described by Konikow and Bredehoeft (1978).

In summary, the governing equation describing two-dimensional horizontal ground-water flow can be written (Konikow and Grove,1977)

S a _agT T a =

(Hs - h) - Q xx 5 yy (A-1) in which S[-] = the storage coefficient h[L] = the hydraulic head xx and Tyy[L2/T]

T

= the aquifer transmissivities in the x and y directions NUREG-1101, Vol. 3 A-1 R

K [L/T] = the vertical hydraulic conductivity of an adjacent z

low permeability layer, streambed, or lakebed m[L] = the thickness of the adjacent low permeability layer, streambed, or lakebed Hs [L] = the hydraulic head in the source (sink) bed, stream, or lake Q[L/T] = the rate of withdrawal (+) or recharge (-)

x .nd y[L] = the Cartesian coordinates aligned with the principal axes of the transmissivity tensor Transmissivity is defined as the hydraulic conductivity integrated over.the saturated thickness (Bear, 1979). If hydraulic conductivity is considered con-stant over the saturated thickness T

xx

• K xx b (A-2)

T =K b yy in which K and Kyy[L/T] = the, hydraulic conductivities in the x and y directions xx (principal axes) b[L] = the saturated thickness For unconfined or phreatic aquifers, which are not considered here, transmis-sivity is a function of hydraulic head (water table elevation). The terms K 7 a.1d m relate to leakage to or from adjacent aquifers or water bodies,'as well as boundary conditions (see Figure A-1).

The model's numerical procedures allow simulation of aquifers with spatially variable T, K7/m, Hs , and'Q. The storage coefficient 5 is uniform over the entire aquifer. In addition, transmissivity is not a function of hydraulic head; thus equation A-1 represents a confined or linearized aquifer.

Flow equation A-1 is solved for hydraulic head h(x, y, t), using a rectangular, uniformly spaced, block-centered, finite difference grid (cf., Konikow and Bredehoeft, 1978; Pinder and Bredehoeft, 1968). From this solution at each time step, velocities Vxand Vy[L/T] required for transport calculations are calculated from finite difference approximations to Darcy flux K

xx ah V

x .n ax (A-3)

V

- hnah5y y

where n[L3/L3] = porosity NUREG-1101, Vol. 3 A-2 d

l

/ \

Adjacent Aquiter a

//4N M/

gelativeIY N gmperViOuS 3

Y Unit \

h5 '  %. #O

'e .g* ,

N pumping 2 p gN t ih g went tow PermeabilitV y,, s SUga Of

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~,i o fer o  ; -

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h j i j  ;

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h } ncon, fined.

1 Aquiter j e i i Hs lhl/, I '

ll NY l h // ll &,, f //'//

OATUM Figure A-1 Definition sketch of horizontal groundwater flow t

l 1

HOREG-1101. VOI* 3 A-3 l

The velocity calculations are crucial to the accuracy of the transport solution and important updates to the original program include velocity determination near aquifer boundaries (cf., Appendix B).

RADIONUCLIDE TRANSPORT SIMULATION Tracy (1982) modified the Konikow-Bredehoeft model to incorporate retardation from sorption and radioactive decay. In addition, a minor error in the govern-ing equation development by Konikow and Grove (1977) was pointed out (cf.,

Konikow and Grove, 1984).

l The governing equation describing two-dimensional horizontal radionuclide trans-port can be written (Konikow and Bredehoeft, 1978) a, a Rg aq bD j )a a ~Y x -V y -R d AC + f(C-C') (A-4)

I in which 4

l Rd [-] = the retardation coefficient C[M/'.3] = radionuclide concentration

b[L] = aquifer saturated thickness

D jj[L2/T] = the dispersion coefficient ,

W (L/T] = all recharge terms (only if W is negative)

C'[M/L8] = the cor. centration in recharge W_

A[T 1] = the radioactive decay constant and subscripts i, j represent the coordinate axes: 1, j = 1 corresponding to x, and i, j = 2 corresponding to y. All parameters in equation A-4, except poros-ity n may vary in space as specified by the modeler.

I The recharge summation in equation A-4 represents distributed recharge W_=Q, leakage W_ = -K (H - h)/m, and injection of water into the aquifer. The cor-7 3 responding concentration in the source water C' is used for each different source.

t Tracy (1982) describes the incorporation of decay in the model's numerical solution through an explicit scheme which averages the exponential concentration change over each time step.

The Dispersion Term The first term on the righthand side of equation A-4 represents dispersion and can be written .

E\ ay g yyay/ E\

4 i 1a 3C =1 .a [bD E -- E ay(yxax/+0[bD bD xyBy\+0[bD - --

(A-5) b ax j bD ij 0x) b ax ( xx ax/ + 0ax g j The dispersion coefficients Dj ) are evaluated from the logitudinal and trans- l Verse dispersion coefficients D Land D Talong the flowpath and normal NUREG-1101, Vol. 3 A-4 l

(perpendicular) to the flowpath, respectively, which are assumed to be propor-tional to the bulk velocity magnitude V DL * "LV (A-6)

DT " "TV in which ag [L] = longitudinal dispersivity aT[L] = transverse dispersivity V= Vx +V2 y

The directional terms in equation A-5, for an isotropic medium, are then (Bear, 1979; Konikow and Bredehoeft, 1978) y2  ;

D xx

=a'L V*2 V +a T V E- l j

i V2 y2 l yy =a l D (A-7)

• "T

/ 3 VV xy xy =D yx O

V

• ("L ~ "T If the flow occurs only in the x direction, V =y 0, and these terms reduce to O

xx * "L x D

yy =aVTx (A-8)

D xy

=Dyx =0 The Retardation Term i

A constant retardation Rd accounts for solute mass which is sorbed to the porous medium according to a linear reversible equilibrium sorption law (Bear, 1979,

p. 242). Tracy (1982) has modified the Konikow-Bredehoeft model to include this retardation, and to approximate non-constant retardation due to nonlinear i equilibrium sorption.

l l If sorbed concentration C s

[M/M] is linearly proportional to solute concentra-l tion in groundwater, the change of sorbed mass storage can be written dC dt s=g E M-9) d at

( in which K d [L /M] is the distribution coefficient. In a unit volume of a saturated aquifer, the total change in mass storage is the change in sorbed NUREG-1101, Vol. 3 A-5

concentration times the aquifer matrix mass per unit volume, plus the change in dissolved concentration times the porosity ps (1 ~ ") t

• " " Ps(1 - n) Kd +"

/ p (1-n)Kdi aC (A-10)

=n 1+ s n f 5t

= nR d5 in which ps [M/L ] = the aquifer matrix solid density Rd

• 1 + Ps (1 - n)Kd" Thus, the lefthand side term in equation A-4, R d , accounts for mass storage both in solution and on the aquifer matrix (normalized by porosity).

In general, sorption is not exactly linear with concentration. Retardation resultingfromnonlinearsorptioncanbeapproximatedbycomputingRywhichis dependent on concentration C. This approximation becomes more accurate as the equilibrium relationship over the applicable range approaches linearity.

The nonlinear Freundlich equilibrium isotherm is N

Cs =KCf (A-11) in which K [L8/M] corresponds to Kd above, and N[-] is the Freundlich exponent.

f The approximate retardation factor is (Tracy, 1982) ps (1 - n) N-1 Ry = 1 + K NC (A-12) n f which is a function of solute concentration C.

The nonlinear Langmuir equilibrium isotherm is KgC'C C, (^ )

= (1 + K gC) in which Kg[L 8/M] corresponds to K ;d and C'[M/M] is the maximum sorbed con-centration. At low concentrations Cs = K C'C as C+0 (A-13a) g i

NUREG-1101, Vol. 3 A-6

At high concentrations CsyC;asC+= (A-13b) and sorbed concentration is essentially constant. The corresponding retardation factor is (Tracy,1982) p (1 - n) s KC gs (A-14)

Rd*1* n (1 + K gC)z which depends on solute concentration C.

To calculate the nonlinear retardation factors for a new particle move, the concentrations at the last particle move are used for C. All other parameters in equations A-12 and A-14 are constant. This explicit formulation loses accuracy as the concentration change between particle moves increases.

NOTES ON MODEL APPLICATION Numerical solutions of the flow and solute transport equations involve approxi-mations which are necessary in order to solve complex non-homogeneous field problems in a cost-effective manner. To ensure reasonable accuracy, the modeler must follow the application procedures outlined by Konikow and Bredehoeft (1978) and Tracy (1982). A discussion of related model behavior is presented below.

This discussion centers on the numerical solution of the solute transport equa-tion; Konikow and Bredehoeft (1978) and Pinder and Bredehoeft (1968) describe the application of the finite difference flow model.

Initial and Boundary Conditions Initial and boundary conditions, that is, concentrations and solute mass flux rates, must be specified for a solution of the radionuclide transport equation.

The initial concentration in the aquifer may represent existing contamination at the beginning of the simulation period, or " background" concentration. In the latter case, it should be noted that the initial background concentrations will decrease, because of radioactive decay, in the absence of a replenishing source. In both cases, specification of an initial concentration in solution directly implies a sorbed concentration governed by the appropriate linear or nonlinear equilibrium' law.

The boundary conditions for the transport simulation are derived directly from the boundary conditions for the flow equation. The entire flow domain is surr rounded by a no-flow boundary. Within this boundary, nodes can be specified as a fixed flux boundary, in which the discharge of flow, but not the direction, is specified, or as a third-type leaky boundary. The numerics of the third-type boundary are identical to calculation of leakage to the aquifer from an overlying water table aquifer or lake. By specifying a very high leakage term, Kz /m, the head in the aquifer is essentially constant and equal to the head in the leakage source. In this way, the leakage term in the governing flow equa-tion A-4 can be used to epproximate a fixed-head boundary. Unless a fixed flux or leakage (or fixed head) condition is specified at the model edge, the model assumes that the edge of the domain is a no-flow boundary.

NUREG-1101, Vol. 3 A-7

Radionuclide concentrations must be specified in the fixed flux and leakage boundaries which act as sources of flow. When these boundary conditions act as sinks, the concentration in the flux out of the domain is the concentration at that node. In this sense, the solute boundary condition is simply a specified advection of solute mass in the flowing groundwater. No dispersion occurs at these boundaries. In addition, it is not possible to specify a fixed concen-tration of solute at a node in this model (Konikow and Bredehoeft, 1978, p. 35).

The assumption of no dispersion at the aquifer boundary must be considered when evaluating model results. In the model, solute n. ass will not disperse across a no-flow boundary which represents a flow streamline. In reality, solute mass moves across a streamline because of lateral dispersion. Ignoring this effect will increase concentrations near the boundary. In addition, no solute mass can disperse upstream through a flow boundary source. Neglecting this disper-sion can have a significant impact on accuracy if a radionuclide source is located near an upgradient boundary. To preclude these effects, flow bound-aries and radionuclide sources should be so located that radionuclide concen-trations are small at flow source and at no-flow boundaries. Two exceptions to this guideline are (1) a no-flow boundary which is a line of symmetry (i.e., a plume centerline) for the radionuclide transport problem, and (2) a no-flow boundary which represents a geologic boundary (fault or pinchout) for both flow

, and transport.

Model Accuracy The accuracy of this model is evaluated through computations by the computer program and application of the judgment and experience of the modeler. The model's accuracy is partially a function of the finite difference grid, loca-tion of sink / sources and boundary conditions (discussed above), and model execution parameters.

The accuracy of finite difference type calculations and the moving particle technique employed by the model are largely determined by grid spacing and time step size. The model automatically sets time step size based on stability criteria and the execution parameters discussed below. For accurate finite difference calculations of radionuclide dispersion, the change in concentration between nodes murt be relatively small. Radionuclide transport includes decay which can greatly reduce concentrations between nodes, depending on the retarded velocity. A change in system parameters such as velocity or retardation may require a major change in the model grid. One procedure for checking the effect of grid size on model accuracy is to reduce grid spacing until the change in model output from one grid size to the next is insignificant. Unfortunately, this procedure may have to be repeated for each set of system parameters, especially if velocity, retardation, or decay rate are significantly changed.

In addition, changes in grid block dimensions alter the model representation of the source area. For example if both x and y grid dimensions are halved, it may be necessary to represent sources with four times as many blocks.

Konikow and Bredehoeft (1978) describe the control of mass balance error through the two execution parameters: NPTPND, the initial number of particles per node; and CELDIS, the maximum fraction of grid dimensions that the particles are allowed to move in one step. The model computes the mass balance error for flow and radionuclide at certain time steps. The cumulative radionuclide mass balance error is the relative difference between the mass flux through the system (total in minus total out) and the change in total mass stored in the NUREG-1101, Vol. 3 A-8

system (current storage minus initial storage). M0CMOD also accounts for l adsorbed radionuclide mass storage and decay. This measure does not necessarily ensure accuracy, but it is a check on the global balance of the model and a

good indicator of overall consistency. NPTPND should be specified at its maximum value, 9, for minimum error. The optimum value of CELDIS is problem specific and should be determined through experimentation and modeler judgment.

Konikow and Bredehoeft (1978) recommend CELDIS = 0.5 as a general guide for minimum error. Table A-1 shows the variation of mass balance error for a uniform flow field problem for which CELDIS = 0.3 produced minimum error. It appears that numerical errors compound in a uniform flow field problem in which all particles are behaving identically. Thus the mass balance may be consider-ably poorer than that for a non-uniform flow field in which errors at different nodes tend to offset each other. Table A-1 also shows that changes in computed concentration are reflected in the changes in mass balance error.

Table A-1 Effects of CELDIS on mass balance error and computed concentration for a uniform flow field problem b Mass c

a Time Number of balance error Croo CELDIS (years) particle moves (percent) (mg/ liter) 0.2 0.55 20 -8.75 20.9 0.2 2.19 80 -7.53 21.7 0.2 10.0 366 -7.21 20.2 0.25 0.68 20 -8.44 21.1 0.25 2.05 60 -6.58 21.3 0.25 10.0 293 -6.10 20.5 4

0.3 0.82 20 -3.59 20.1 0.3 2.46 60 -1.51 19.9 0.3 10.0 244 -1.35 18.7 0.49 1. 3 20 7.21 18.2 0.49 2.6 40 8.88 14.7 0.49 9.3 140 7.40 19.0 0.5 1.36 20 12.0 15.5 0.5 2.72 40 13.9 15.5 0.5 10.0 147 10.7 15.4 0.75 2.04 20 20.4 13.0 0.75 10.0 98 24.8 14.1 a

CELDIS is the maximum fraction of grid dimensions that the particles are allowed to move in one step.

b The entire system reaches steady state after about 2 years.

c C oo is the computed concentration 100 ft (2 nodes) directly i

downgradient from the injection point. Velocity is 1 ft/ day.

1 NUREG-1101, Vol. 3 A-9

SU M RY A numerical model for radionuclide transport in groundwater has been developed (Konikow and Bredehoeft, 1978; Tracy, 1982) which simulates two-dimensional hori-zontal flow and transport of a decaying radionuclide with retardation from adsorption and both longitudinal and transverse dispersion. The computer model M0CM0084 can simulate groundwater systems with heterogeneous properties, complex geometries, and non-uniform source conditions. The model is applicable to both steady-state and transient flow conditions. Responsibility for model accuracy rests on the modeler who can use computed mass balance errors to refine grid spacing and execution parameter selection. Properly used, the model can provide an accurate estimate of potential radionuclide transport in groundwater systems.

REFERENCES

. Bear, J., Hydraulics of Groundwater, McGraw-Hill, New York, 1979.

Konikow, L. F., and J. D. Bredehoeft, " Computer Model of Two-dimensional Solute Transport and Dispersion in Ground-water," U.S. Geological Survey Techniques of Water Resources Investigations, Book 7, Chapter C2, 1978.

Konikow, L. F., and D. B. Grove, " Derivation of Equations Describing Solute, Transport in Groundwater," U.S. Geological Survey Water Resources Investigation 77-19, 1977, Revised 1984.

Pinder, G. F. , and J. D. Bredehoeft, " Application of the Digital Computer for Aquifer Evaluation," Water Resources Research, 4(5): 1069-1093, 1968.

Tracy, J. V., " Users Guide and Documentation for Adsorption and Decay Modifica-tions to the USGS Solute Transport Model," U.S. Nuclear Regulatory Commission, NUREG/CR-2502, 1982.

NUREG-1101, Vol. 3 A-10

APPENDIX B OCTOBER 12, 1983 UPDATE TO THE USGS SOLUTE TRANSPORT MODEL NUREG-1101, Vol. 3

NOTE ON COMPlTIT.R PROGRAM UPDATE October 12, 1983

Reference:

" Computer model of two-dimensional solute transport and dispersion in ground-water," by L. F. Konikow and J. D. Bredehoeft (1978): U.S.

Geological Survey Techniques of Water-Resources Investigations, Book 7, Chapter C2.

I Several additional program modifications have been made since the previous

, update of August 26, 1981. These can be implemented as follows:

1. The following changes assure that all time-step and flow calculations are perfomed entirely in double precision. These changes will only make a difference on certain computers.

I a) Delete the following Fortran statements from the source code:

A 115 B 55 b) insert the following Fortran statements in the proper sequential location:

REAL *8TMSUM,ANTIM,TDEL REAL *8TMSUM,ANTIM,TOEL A 116 B 56 REAL *80XINV,0YINV,ARINV,PORINV S 57 REAL *80XINV,0YINV,ARINV,PORINV C 45 REAL *8DXINV,0YINV,ARINV,PORINV .

E 35 REAL +80XINV,DYINV,ARINV,PORINV F 32 REAL +8DXINV,0YINV,ARINV,PORINV G 35

2. The fo!!owing format changes are made to provide more convenient printouts for most situations.

a) Delete:

81630 B2620 B3603 B2330 B3440 b) Insert:

160 WRITE (6,840) (VPRM(IX,1Y),IX=1,NX) B1631 260 WRITE (6,840) (PERM (Ix,1Y),IX=1,Nx) B2331 320 WRITE (6,840) (VPRM(IX,IY),1Xaj,NX) B2621 l

NUREG-1101, Vol. 3 8-1

3. The following changes will assure that all output routines are called at the end of the last time step of a pumping period.

a) Delete:

A 450 A 610 b) Insert:

IPCK=0 A 336 IF (TOEL.EQ.(PYR-TINT)) IPCK=1 & 406 IF (REMN.EQ.0.0.0R.N.EQ.NTIM.0R.IPCK.EQ.1) CALL OUTPT A 451 120 IF (REMN.EQ.0.0.0R.N.EQ.NTIM.OR. MOD (N,50).EQ.0.0R.IPCK.EQ.1J A 611 1 CALL CWMOT A 612

4. The following changes will assure that an endfile record is written after the last data record. This will only make a difference on certain computers. (No deletions required.)

a) Insert:

ENOFILE(6) A 702 IF (NPNCHV.EQ.0) GO TO 155 A 703 ENOFILE(7) A 704 155 CONTINUE A 705

5. The following changes were made primarily to improve the interpolated estimates of velocities at tracer particles located in wells adjacent to no-flow boundaries and to assure a more consistent and uniform regeneration of particles at fluid sources.

a) Delete:

E 420 F 780 F1660 F1990 F2815 E 430 F 800 F1840 F2715 F2825 E 440 F1010 F1860 F2725 F2835 E 450 F1030 F1880 F2735 F3390 E 500 F1070 F1890 F2745 F3395 E 510 F1280 F1920 F2755 F3410 E 520 F1300 F1930 F2765 F3420 E 530 F1340 F1940 F2775 F3430 F1560 F1950 F2785 F3440 F1580 F1960 F2795 F3450 F1620 F1980 F2805 F3460 2

NUREG-1101, Vol. 3 B-2 l

- - . _ - - n- _. __-y _

b) Insert:

C

• REVISED OCTOBER 1983

• A 58 DMX=HK(Ix-1,IY)-HK(Ix+1,IY) E 421 IF (THCK(IX-1,1Y).EQ.0.0) DMX=HK(1x,IY)-HKCIX+1,IY) E 431 IF (THCKCIX+1,1Y).EQ.0.0) DMX=HK(Ix-1,IV)-HK(Ix,IY) E 441 IF (THCK(IX-1,IV).EQ.0.0.AND.THCK(Ix+1,1Y).EQ.0.0) OHx=0.0 E 451 GR0x=0HX*0x!NV*0.50 E 455 DHY=HKCIX,1Y-1)-HKCIX,IY+1) E 501 IF (THCK(IX,1Y-1).EQ.0.0) DHY= HK(Ix,IY)*HK(IX,IY+1) E 511 IF (THCK(IX,IY+1).EQ.0.0) DHYa MK(IX,IY-1)-HK(IX,IY) E 521-IF (THCK(IX,IY-1).EQ.0.0.AND.THCK(IX,IY+1).EQ.0.0) DHY=0.0 E S31 GROY=DHYaDYINV*0.50 E 535 IC089 F 595 IC0=1 F 705 IF (THCK(1xE+1,IYS).GT.O.0) VXSE=Vx5W F 782 IF (THCK(IXE,IYS+1).GT.O.0) VYSE=VYNE F 802 IC0=2 F 925 IF (THCK(IVX-1,1YS).GT.O.0) VXSw=Vx5E F1012 IF (THCK(IvX,IYS+1).GT.O.0) VYSW=VYNW 51032 IF (THCK(IVX,IYS+1).GT.O.0) VYSW=VYNW F1072 IC0=3 51195 IF (1HCK(IxE+1, IVY).GT.O.0) VXNE=VxNW F1282 IF (THCK(IXE, IVY-1).GT.O.0) VYNE=VYSE F1302 IF (THCK(IXE+1, IVY).GT.O.0) VxNE=VxNW F1342 IC0=4 F1465 IF (THCK(IVX-1, IVY).GT.O.0) VXNW:VxNE 51562 IF (THCK(IVx, IVY-11.GT.O.0) VYNW:VYSW F1582 4

IF (THCK(IVX-1, IVY).GT.O.0) VXNW=VXNE F1622 IF (THCK(IVX, IVY-1).GT.O.0) VYNW=VYSW F1662 C ---

CHECK FOR ADJACENT NO-FLOW BOUNDARIES--- F17414 GO TO (1270,1275,1280,1285,1290) ICO F17418 GO TO 1290 F1741C 1270 IF ( T H C K ( I x E , I V,Y ) . E Q . 0. 0 ) GO TO 1272 F1742a IF (THCK(IVX,IYS).EQ.0.0) GO TC 1273 F17426 IF (THCK(IVX, IVY).EQ.0.0) GO TO 1274 F1742C GO TO '

1290 F17420 1272 VxNE=vXSE F1742E IF (THCKCIvX,IYS).GT.O.0) GO TO 1274 F1742F 1273 VYSWsVYSE F 174 2G 1274 VxNW=VXSW F1742H VYNWaVYNE F1742I GO TO 1290 F1742J l

I 3

NUREG-1101, Vol. 3 B-3

56) Inserts (continued) 1275 IF (THCK(IvXrIVY).EQ.0.0) GO TO 1277 F1744A IF (THCK(IXE,IYS).EQ.0.0) GO TO 1278 F17448 IF (THCK(IXE, IVY).EQ.0.01 GO TO 1279 F1744C GO TO 1290 F17440 1277 VXNW=VXSW F1744E IF (THCKCIXErIYS).GT.O.0) GO TO 1279 F1744F 1278 VYSE=VYSW F1744G 1279 VXNE=VXSE F1744H VYNE=VYNW F17441 GO TO 1290 51744J 1280 IF (THCK(IXE,IYS).EQ.0.0) GO TO 1282 F1746A IF (THCK(IVX, IVY).EQ.0.0) GO TO 1283 F1746B IF (THCK(IVX,IYS).EQ.0.0) GO TO 1284 F1746C GO TO 1290 F17460 1282 VXSE=VXNE F174cE IF (THCK(IVXrIVY).GT.O.0) GO TO 1284 F1746F 1283 VYNW:VYNE F1746G 1284 Vx5w=VXNW F174eH VYSw=vYSE F1746I GO TO 1290 F1746J 1285 IF (THCK(IVX,IYS).EQ.0.0) GO TO 1287 F1748A IF (THCK(IXE, IVY).EQ.0.0) GO TO 1288 F17488 IF (THCK(IXErIYS).EQ.0.0) GO TO 1289 F1748C GO TO 1290 F17480 1287 VXSW=VXNW F1748E IF (THCK(IXErIvY).GT.O.0) GO TO 1289 F1748F 1288 VYNE=VYNW F1748G 1289 VYSE=VYSW F1748H VXSE=VXNE F1748I 1290 CONTINUE F1749A CELYD=CELOYH+2.0 F1921 V Y W 8 V Y NW * ( 1. 0-f E 8. 7 0 ) + V Y $ w er F t. Y D F1931 VYE=VYNEa(1.0-CELYD)+VYSE*CELYO F1951 4

NUREG-1101, Vol. 3 B-4 l _ _ _ _ _ _ _ _ _ _ _ _ - -

i l

)

APPENDIX C LISTING OF M0CM0084 COMPUTER PROGRAM NUREG-1101, Vol. 3

LISTING 0F MOCM0D84 FORTRAN (CDC)

PROGRAM MOC(INPUT,0VTPUT, TAPE 5= INPUT, TAPE 6=0VTPUT. TAPE 7) N 01 C LOW-LEVEL WASTE AND URANIUM REC 0VERY PROJECTS BRANCH N 02 C AND GE0 TECHNICAL BRANCH, DIVISION OF WASTE MANAGEMENT N 03 C U.S. NUCLEAR REGULATORY COMMISSION N 04 C WASHINGTON, DC 20555 N 05 C CHANGES FROM USGS VERSION NOTED BY "N" IN COLUMN 75 N 06 C **************************************************************** A 10 C *

• A 20 C

• C

• SOLUTE TRANSPORT AND DISPERSION IN A POROUS MEDIUM

• A 30 NUMERICAL SOLUTION --- METHOD OF CHARACTERISTICS *

• A 40 C

• PROGRAMMED BY J. D. BREDEH0 EFT AND L. F. K0NIK0W A 50 C REVISED APRIL 1979, MARCH 1980

• C

• A 55 C

• REVISED DECEMBER 1980

• A 56 C

• REVISED AUGUST 1981

• A 57 C

• RADIONUCLIDE DECAY AND NONLINEAR SORBTION MODIFICATIONS

• N 10 C

• DEVELOPED AND IMPLEMENTED FOR US NRC BY J. TRACY

• N 20 SEE TRACY, J.V. USERS GUIDE AND DOCUMENTATION FOR

• N 22 C

• ADSORPTION AND DECAY MODIFICATIONS TO THE USGS

• N 23 C

• C

• SOLUTE TRANSPORT MODEL. U.S. NUCLEAR REGULATORY

• N 24 COMMISSION, NUREG/CR-2502, WASHINGTON, DC. 1982. ~* N 25 C

• C

• MODIFIED AUGUST-SEPTEMBER 1981

• N 30 REVISED OCTOBER 1983

• A 58 C *

• A 60 C **************************************************************** A 70 C* IBM REAL*8 TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,AOPT, TITLE A 90 C* IBM REAL*8 XDEL,YDEL,S, AREA,SUMT,RH0,PARAM, TEST,TOL, PINT,HMIN,PYR A 100 C* IBM REAL*8 TINT, ALPHA 1,ANITP A 110 C* IBM REAL*8 TMSUM,ANTIM,TDEL A 116 LEVEL 2,N0DEID,NPCELL,NP0LD LIMB 0,IX0BS.IYOBS.THCK, PERM N 32 LEVEL 2,TMWL,TM0BS,TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM N 33 LEVEL 2,A0PT, TITLE,PART, CONC,TMCN,VX,VY,CONINT,CNRECH N 34 LEVEL 2,VXNORD.VYNORD l N 35 COMMON /PRMI/ NTIM,NPMP,NPNT,NITP,N,NX,NY,NP.NREC, INT,NNX,NNY,NUM0 A 120 1BS,NMOV IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N A 130 2PNCHV,NPDELC,ICHK A 141 COMMON /PRMC/ N0DEID(42,42),NPCELL(42,42),NP0LD(42,42), LIMB 0(500), A 145 IIX0BS(10),1YOBS(10) 2,NDECAY,NSORB A 146 N 40 COMMON /HEDA/ THCK(42,42), PERM (42,42),TMWL(10,50),TMOBS(50),ANFCTR A 170 COMMON /HEDB/THRX(42,42,2),VPRM(42,42),HI(42,42),HR(42,42),HC(42, A 180 142),HK(42,42),WT(42,42), REC (42,42),RECH(42,42), TIM (100),A0PT(42),T A 190 l

l l

NUREG-1101, Vol. 3 C-1

21TLE(10),XDEL,YDEL,S. AREA,SUMT, RHO,PARAM, TEST,TOL, PINT,HMIN,PYR A 200 COMMON /CHMA/ PART(3,9000), CONC (42,42),TMCN(10,50),VX(42,42),VY(42 A 210 1,42),CONINT(42,42),CNRECH(42,42),POROS,SUMTCH, BETA,TIMV, STORM,STOR A 220 2MI,CMSIN. CMS 0VT,FLMIN FLMOT,SUMIO.CELDIS,DLTRAT,CSTORM A 230 3,DCYLAM,BLKDEN,SRBRAT,SRBSAT,SRBALF,VOLOCY,VOLSRB,SRBDCY N 50

' COMMON /VNORD/ VXNORD(42,42),VYNORD(42,42) N 55 C *************************************************************** A 240 C ---LOAD DATA--- A 250 INT =0 A 260 TMSUM = 0.0 A 265 CALL PARLOD A 270 CALL GENPT A 280 C *************************************************************** A 290 i C ---START COMPUTATIONS--- A 300 C ---COMPUTE ONE PUMPING PERIOD--- A 310 00 150 INT =1,NPMP A 320 A 325 IF(INT.GT.1)TMSUM=TMSUM+PYR A 330 IF (INT.GT.1) CALL PARLOD IPCK=0 A 336 C ---COMPUTE ONE TIME STEP--- A 340 D0 130 N=1,NTIM A 350 IPRNT=0 A 360 C ---LOAD NEW DELTA T--- A 370 TINT =SUMT-TMSUM A 381 TDEL= AMIN 1(TIM (N),PYR-TINT) A 390 SUMT=SUMT+TDEL A 400 A 406 IF(TDEL.EQ.(PYR-TINT))IPCK=1 A 410 TIM (N)=TDEL REMN= MOD (N,NPNT) A 420 C *************************************************************** A 430 A 435 IF (S.EQ.0.0.AND.ICHK.EQ.0.AND.(N.GT.1.0R. INT.GT.1)) GO TO 101 A 440 CALL ITERAT A 451 IF (REMN.EQ.0.0.0R.N.EQ.NTIM.0R.IPCK.EQ.1) CALL OUTPT A 460 CALL VELO A 471 101 CALL MOVE C *************************************************************** A 480

---STORE OBS. WELL DATA FOR TRANSIENT FLOW PROBLEMS---

A 490 C

A 500 IF (S.EQ.0.0) GO TO 120 A 510 IF (NUM0BS.LE.0) GO TO 120 A 520 J= MOD (N,50)

A 530 IF (J.EQ.0) J=50 A 540 TMOBS(J)=SUMT 00 110 !=1,NUMOBS A 550 NUREG-1101, Vol. 3 C-2

h TMWL(I,J)=HK(IX0BS(I),IYOBS(I)) , A 560 TMCN(IJ)= CONC (IX0BS(I),1Y0BS(I)) A 570 110 CONTINUE A 580 C *************************************************************** A 590 C ---0UTPUT ROUTINES--- A 600 120IF(REMN.EQ.0.0.0R.N.EQ.NTIM.0R.M0D(N,50).EQ.0.0R.IPCK.EQ.1) A 611 1 CALL CHMOT A 612 IF (SUMT.GE.(PYR+TMSUM)) GO TO 140 A 621 130 CONTINUE A 630 C *************************************************************** A 640 C ---

SUMMARY

OUTPUT--- A 650 140 CONTINUE A 660 IPRNT=1 A 670 CALL CHM 0T A 680 150 CONTINUE A 690 C *************************************************************** A 700 C* IBM ENDFILE(6) A 702 C* IBM IF (NPNCHV.EQ.0) GO TO 155 'A 703 C* IBM ENDFILE(7) A 704 155 CONTINUE A 705 STOP A 710 C **************************=************************************ A 720 END A 730-SUBROUTINE PARL0D B 10 C* IBM DOUBLE PRECISION DMIN1.DEXP,DLOG, DABS B 20 C* IBM REAL*8 TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT, TITLE B 30 C* I BM REAL*8 X DEL ,Y D EL ,S , AREA , SUMT ,RH0, PA RAM ,TE ST ,TOL , P I NT ,HMI N , PY R B 40 C* IBM REAL*8 FCTR,TIMX,TINIT, PIES,YNS,XNS, RAT.HMX,HMY B 50 C* IBM REAL*8 TMSUM,ANTIM,TDEL B 56 C* IBM REAL *8DXINV,DYINV,ARINV,PORINV B 57 C* IBM REAL*8 TINT, ALPHA 1,ANITP B 60 INTEGER OVERRD B 65 LEVEL 2,N0DEID,NPCELL,NPOLD, LIMB 0,IXOBS.IY0BS,THCK PERM,TMWL,TMOBS- N 52 LEVEL 2,TMRX,VPRM HI,HR.HC HK,WT, REC,RECH, TIM,A0PT, TITLE,XDEL,YDEL N 53 LEVEL 2,PART, CONC,TMCN,VX,VY,CONINT,CNRECH,SUMC.VXBDY,VYBDY N 54 LEVEL 2,VXNORD,VYNORD , N 55 COMMON /PRMI/ NTIM,NPMP.NPNT,NITP,N,NX,NY,NP,NREC, INT.NNX,NNY,NUM0 8 70 IBS.NMOV,IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N B 80 2PNCHV,NPDELC,1CHK B 91 COMMON /PRMC/ N0DEID(42,42),NPCELL(42,42),NPOLD(42,42), LIMB 0(500), B 95 IIX0BS(10) IY0BS(10) B 96 2,NDECAY,NSORB N 60 COMMON /HEDA/ THCK(42,42), PERM (42,42),TMWL(10,50),TM0BS(50),ANFCTR B 120

/

1 NUREG-1101, Vol. 3 C-3

COMMON /HEDB/ TMRX(42,42.2),VPRM(42,42),HI(42,42).HR(42,42),HC(42, B 130 142),HK(42,42),WT(42,42), REC (42.42),RECH(42,42), TIM (100),A0PT(42),T B 140 2ITLE(10),XDEL,YDEL S, AREA,SUMT, RHO,PARAM. TEST,TOL, PINT,HMIN,PYR B 150 COMON /CHMA/ PART(3,9000), CONC (42,42),TMCN(10,50),VX(42,42),VY(42 B 160 1,42),CONINT(42,42).CNRECH(42,42),POROS,5UMTCH. BETA.TIMV, STORM,STOR B 170 2MI.CMSIN.CMSOUT,FLMIN,FLM0T SUMIO.CELDIS,DLTRAT,CSTORM B 180 3,DCYLAM,BLKDEN,SRBRAT,SRBSAT,SRBALF,VOLDCY,VOLSRB,SRBDCY N 70 COMMON /VNORD/ VXNORD(42,42),VYNORD(42,42) N 75 COMMON / BALM / TOTLQ,TOTLQI.TPIN,TP0UT 3 186 COMMON /XINV/ DXINV,0YINV,ARINV,PORINV B 200 COMMON /CHMC/ SUMC(42,42),VXBDY(42,42),VYBDY(42,42) B 210 C *************************************************************** 'B 220 IF (INT.GT.1) GO TO 10 B 230 WRITE (6.750) B 240 !

READ (5,720) flTLE B 250 l WRITE (6.730) TITLE B 260 C *************************************************************** B 270 C ---INITIALIZE TEST AND CONTROL VARIABLES--- B 280 STORMI=0.0 B 290 TEST =0.0 B 300 T0TLQ=0.0 B 310 T0TLQI=0.0 B 315 TPIN = 0.0 B 317 TPOUT = 0.0 B 318 SUMT=0.0 B 320 SUMTCH=0.0 B 330 INT =0 B 340 IPRNT=0 B 350 NCA=0 B 360 N=0 B 370 IM0V=0 B 380 NM0V=0 B 390 ICHK = 0 8 395 V0LDCY=0.0 N 80 VOLSRB=0.0 N 90 SRBDCY = 0.0 N 100 C *************************************************************** B 400 C ---LOAD CONTROL PARAMETERS--- B 410 READ (5,740) NTIM,NPMP,NX,NY,NPMAX,NPNT,NITP,NUMOBS,ITMAX,NREC,NPT B 420 1PND NC0 DES.NPNTMV,NPNTVL,NPNTD,NPDELC,NPNCHV B 430 READ (5,800) PINT,TOL,POROS, BETA,S,TIMX,TINIT,XDEL,YDEL,DLTRAT,CEL B 440 1 DIS,ANFCTR B 450 READ (5,805) NDECAY,NSORB,0CYTIM.DENROC,SORBQR,SORBST,SORBAL N 110 NUREG-1101, Vol. 3 C-4

NNX=NX-1 B 470 NNY=NY-1 B 480 NP=NPMAX B 490 DXINV=1.0/XDEL B 500 DYINV=1.0/YDEL B 510 ARINV=DXINV*DYINV B 520 PORINV=1.0/POROS B 530 DCYLAM = 0.0 N 120 BLKDEN = 0.0 N 130 SRBRAT = 0.0 N 140 SRBSAT = 0.0 N 150 SRBAL F = 0.0 N 160 l IF (NDECAY.GT.O.AND.DCYTIM.GT.O.0) N 170 1 DCYLAM = ALOG(2.0)/(365.25*86400.0*DCYTIM) N 180 IF (NS0RB.GT.0) BLKDEN = DENROC*(1.0-POROS)/FOROS N 190 l

IF (NSORB.GT.0 SRBRAT = SORBQR N 200 IF(NS0RB.EQ.2 SRBSAT = SORBST N 210 IF (NSORB.EQ.3 SRBALF = SORBAL N 220 C ---PRINT CONTROL PARAMETERS--- B 540 WRITE (6.760) B 550 WRITE (6.770) NX,NY,XDEL,YDEL B 560 WRITE (6,780) NTIM,NPMP, PINT,TIMX,TINIT B 570 WRITE (6,790) S.POROS, BETA,DLTRAT,ANFCTR B 580 IF (NDECAY.LT.1) WRITE (6,791) N 230 IF (NDECAY.GT.0) WRITE (6,792) DCYTIM,DCYLAM N 240 IF NSORB.LT.1) WRITE (6,793) N 250 IF NS0RB.GT.0 WRITE (6,794) DENROC,BLKDEN N 260 IF NSORB.EQ.1 WRITE (6,795) SRBRAT N 270 IF NS0RB.EQ.2 WRITE (6,796) SRBRAT,SRBSAT N 280 IF (NSORB.EQ.3) WRITE (6,797) SRBRAT,SRBALF N 290 WRITE (6,870) NITP,TOL,ITMAX,CELDIS,NPMAX,NPTPND B 590 IF(NPTPND.LT.4.0R.NPTPND.GT.9.0R.NPTPND.EQ.6.0R.NPTPND.EQ.7) WRIT B 600 1E (6,880) B 610 IF (NITP.LT.1) WRITE (6,885) B 615 WRITE (6,890) NPNT,NPNT?iV,NPNTVL,NPNTD,NUM0BS,NREC,NC0 DES,NPNCHV,N B 620 1PDELC B 630 GO TO 20 B 650 C *************************************************************** B 660

! C ---READ DATA TO REVISE TIME STEPS AND STRESSES FOR SUBSEQUENT B 670 C PUMPING PERIODS--- B 680 10 READ (5,1060) ICHK B 690 IF ICHK.LT.1) WRITE (6,1110) INT B 695 IF ICHK.LT.1) GO TO 20 B 701 l

l NUREG-1101, Vol. 3 C-5 l l _ _ _ _ _ _ _ _ _ _ _ _ _

READ (5,1070)NTIM,NPNT,NITP,ITMAX,NREC,NPNTMV,NPNTVL,NPNTD,NPDELC B 710 1,NPNCHV, PINT,TIMX,TINIT B 720 WRITE (6,1080) INT B 730 WRITE (6.1090) NTIM.NPNT,NITP,ITMAX,NREC,NPNTMV,NPNTVL,NPNTD.NPDEL B 740 1C,NPNCHV. PINT,TIMX,TINIT B 750 C *************************************************************** B 760 C ---LIST TIME INCREMENTS--- B-770 20 DO 30 J=1,100 B 780 TIM (J)=0.0 B 790 30 CONTINUE B 800 PYR = PINT *86400.0*365.25 B 805 TIM (1)=TINIT B 810 IF (NPNTMV.EQ.0) NPNTMV = 999 8 815 IF (S.EQ.0.0) GO TO 50 B 820 DO 40 K=2 NTIM B 830 40 TIM (K)=TIMX* TIM (K-1) B 840 WRITE (6.470) B 850 WRITE (6,490) TIM B 860 IF (TINIT.GT.PYR) WRITE (6,475) B 865 G0 f0 60 B 870 50 ANTIM = NTIM B 882 DO 55 K=1,NTIM B 884 55 TIM (K)=PYR/ANTIM B 886 WRI1E (6,480) TIM (1) B 890 C *************************************************************** B 900 C ---INITIALIZE MATRICES--- B 910 60 IF (INT.GT.1) GO TO 100 B 920 DO 70 IY=1,NY B 930 DO 70 IX=1,NX B 940 VPRM(IX,1Y)=0.0 B 950 PERMIX,IY)=0.0 8 960 THCK IX,IY)=0.0 B 970 RECH IX,IY)=0.0 B 980 CNRECH(IX,1Y)=0.0 B 990 RFC'IX,IY)=0.0 B1000 i iiODE1)(IX,IY)=0 B1010 TMRX(IX,1Y,1)=0.0 B1020 TMRX(IX,1Y,2)=0.0 B1030 HI(IX,IY)=0.0 B1040 HR(IX,IY)=0.0 B1050 HC(IX,IY)=0.0 81060 HK(IX,IY)=0.0 B1070 WT(IX,1Y)=0.0 B1080 NUREG-1101, Vol. 3 C-6

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l VX(IX,IY)=0.0 81090 VY(IX,IY)=0.0 B1100 VXBDY(IX,1Y)=0.0 B1110 VYBDY(IX,1Y)=0.0 B1120 VXNORD(IX,1Y)=0.0 N 295 VYNORD(IX,1Y)=0.0 N 296 CONC (IX,1Y)=0.0 B1130 CONINT(IX,1Y)=0.0 B1140 SUMC(IX,IY)=0.0 B1150 70 CONTINUE B1160 C **************************************************************** B1170 C ---READ OBSERVATION WELL LOCATIONS--- B1180 IF (NUMOBS.LE.0) GO TO 100 01190 WRITE (6,900) B1200 DO 80 J=1,NUM0BS B1210 READ (5,700)IX,1Y B1220 WRITE (6,810)J,1X,1Y B1230 IX0BS(J)=IX B1240 80 IYOBS(J)=IY B1250 DO 90 I=1,NUMOBS B1260 DO 90 J=1,50 B1270 TMWL(I,J)=0.0 B1280 90 TMCN(I,J)=0.0 B1290 C **************************************************************** B1300 C

---READ PUMPAGE DATA -- (X-Y C0 ORDINATES AND RATE IN CFS)--- B1310 C ---SIGNS : WITHDRAWAL = P0S.; INJECTION = NEG.--- B1320 C ---IF INJ. WELL, ALSO READ CONCENTRATION OF INJECTED WATER--- B1330 100 IF (NREC.LE.0) GO TO 120 B1340 IF (INT.GT.1.AND.ICHK.LE.0) RETURN B1345 WRITE (6,910) B1350 D0 110 I=1,NREC B1360 READ (5,710) IX,IY,FCTR,CNREC B1370 IF(FCTR.LT.0.0)CNRECH(IX,IY)=CNREC B1380 REC (IX,IY)=FCTR B1390 110 WRITE (6,820) IX,IY, REC (IX,IY),CNRECH(IX,IY) B1400 C **************************************************************** B1410 120 IF (INT.GT.1) RETURN B1420 AREA =XDEL*YDEL B1430 WRITE (6,690) AREA B1440 WRITE (6,600) B1450 WRITE (6,610) XDEL B1460 WRITE (6,610) YDEL B1470 C **************************************************************** B1480 NUREG-1101, Vol. 3 C-7 t.

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C ---READ TRANSMISSIVITY IN FT**2/SEC INTO VPRM ARRAY--- B1490 C ---FCTR = TRANSMISSIVITY MULTIPLIER ---

- FT**2/SEC--- B1500 WRITE (6,530) B1510 READ (5,550) INPUT,FCTR B1520 D0 160 IY=1,NY B1530 IF (INPUT.EQ.1) READ (5,560) (VPRM(IX,1Y),1X=1,NX) B1540 00 150 IXwl,NX B1550 IF (INPUT.NE.1) GO TO 130 B1560 VPRM(IX,1Y)=VPRM(IX,IY)*FCTR B1570 GO TO 140 B1580 130VPRM(IX,1Y)=FCTR B1590 140IF(IX.EQ.1.0R.IX.EQ.NX)VPRM(IX,1Y)=0.0 B1600 IF(IY.EQ.1.0R.IY.EQ.NY)VPRM(IX,IY)=0.0 B1610 150 CONTINUE B1620 160 WRITE (6,840) (VPRM(IX,IY),1X=1,NX) B1631 C **************************************************************** B1640 C ---SET UP COEFFICIENT MATRIX --- BLOCK-CENTERED GRID--- B1650 C ---AVEPAGE TRANSMISSIVITY --- HARMONIC MEAN--- B1660 IF (ANFCTR.NE.0.0) GO TO 170 B1670 WRITE (6,1050) B1680 ANFCTR=1.0 B1690 170 PIES =3.1415927*3.1415927/2.0 B1700 YNS=NY*NY B1710 XNS=NX*NX B1720 HMIN=2.0 N 300 00 180 IY=2,NNY B1740 D0 180 IX=2,NNX B1750 IF (VPRM(IX,IY).EQ.0.0) GO TO 180 B1760 TMRX(IX,1Y,1)=2.0*VPRM(IX,IY)*VPRM(IX+1,1Y)/(VPRM(IX,IY)*XDEL+VPRM B1770 1(IX+1,IY)*XDEL) B1780 TMRX(IX,1Y,2)=2.0*VPRM(IX,1Y)*VPRM(IX,1Y+1)/(VPRM(IX,1Y)*YDEL+VPRM B1790 1(IX,1Y+1)*YDEL) B1800 C ---ADJUST COEFFICIENT FOR ANIS 0TROPY--- B1810 TMRX(IX,IY,2)=TMRX(IX,IY,2)*ANFCTR B1820 t C ---COMPUTE MINIMUM ITERATION PARAMETER (HMIN)--- B1830 IF (TMRX(IX,1Y,1).EQ.0.0) GO TO 180 B1840 IF (TMRX(IX,1Y,2).EQ.0.0) GO TO 180 91850 RAT =TMRX(IX,IY,1)*YDEL/(TMRX(IX,IY,2)*XDEL) B1860 HMX= PIES /(XNS*(1.0+ RAT)) B1870 HMY= PIES /(YNS*(1.0+(1.0/ RAT))) B1880 IF (HMX.LT.HMIN t. MIN =HMX B1890 IF(HMY.LT.HMIN HMIN=HMY B1900 NUREG-1101, Vol. 3 C-8

180 CONTINUE B1910 C **************************************************************** B1920 C ---READ AQUIFER THICKNESS--- B1930 WRITE (6,510) B1940 READ (5,550) INPUT,FCTR B1950 D0 210 IY=1,NY 81960 IF(INPUT.EQ.1) READ (5,540)(THCK(IX,1Y),IX=1,NX) B1970 00 200 IX=1,NX B1980 IF (INPUT.NE.1) GO TO 190 B1990 THCK(IX,IY)=THCK(IX,1Y)*FCTR B2000 GO TO 200 B2010 190 IF (VPRM(IX,1Y).NE.0.0) THCK(IX,IY)=FCTR B2020 200 CONTINUE B2030 210 WRITE (6,500) (THCK(IX,IY),IX=1,NX) B2040 C **************************************************************** B2050 C ---READ DIFFUSE RECHARGE AND DISCHARGE--- B2060 WRITE (6.830) B2070 READ (5,550) INPUT,FCTR B2080 00 240 IY=1,NY B2090 IF (INPUT.EQ.1) READ (5,560) (RECH(IX,1Y),1X=1,NX) B2100 00 230 IX=1,NX B2110 IF (INPUT.NE.1) GO TO 220 B2120 RECH(IX,IY)=RECH(IX,IY)*FCTR B2130 GO TO 230 B2140 220 IF (THCK(IX,IY).NE.0.0) RECH(IX,IY)=FCTR B2150 230 CONTINUE B2160 240 WRITE (6,840) (RECH(IX,1Y),1X=1,NX) 82170 C **************************************************************** B2180 C ---COMPUTE PERMEABILITY FROM TRANSMISSIVITY--- B2190 C ---COUNT NO. OF CELLS IN AQUIFER--- B2200 C

---SET NZCRIT = 2( OF THE N0. OF CELLS IN THE AQUIFER--- B2210 00 250 IX=1,NX 82220 D0 250 IY=1,NY B2230 IF (THCK(IX,IY).EQ.0.0) GO TO 250 82240 l PERM (IX,IY)=VPRM(IX,1Y)/THCK(IX,IY) B2250 l NCA=NCA+1 82260 250VPRM(IX,IY)=0.0 B2270 C

B2280 AAQ=NCA* AREA B2290 NZCRIT=(NCA+25)/50 B2300 WRITE (6,620) 82310 00 260 IY=1,NY B2320 260 WRITE (6,840)(PERM (IX,1Y),IX=1,NX) B2331 NUREG-1101, Vol. 3 C-9

WRITE (6,630) NCA,AAQ,NZCRIT B2340 C **************************************************************** B2350 C ---READ N0DE IDENTIFICATION CARDS--- B2360 C ---SET VERT. PERM., SOURCE CONC., AND DIFFUSE RECHARGE--- B2370 C ---SPECIFY CODES TO FIT YOUR NEEDS--- B2380 WRITE (6,570) B2390 READ (5,550) INPUT,FCTR B2400 D0 280 IY=1,NY B2410 IF (INPUT.EQ.1) READ (5,640) (N0DEID(IX,IY),IX=1,NX) B2420 00 270 IX=1,NX B2430 270 IF (INPUT.NE.1.AND.THCK(IX,IY).NE.0.0) NODEID(IX,IY)=FCTR B2440 280 WRITE (6,580)(N0DEID(IX,1Y),1X=1,NX) B2450 WRITE (6,920) NC0 DES B2460 IF (NC0 DES.LE.0) GO TO 310 B2470 WRITE (6,930) B2480 DO 300 IJ=1,NCODES B2490 1 READ (5,850) ICODE,FCTR1,FCTR2,FCTR3,0VERRD B2500 D0 290 IX=1,NX B2510 D0 290 IY=1,NY B2520 IF (N0DEID(IX,1Y).NE.ICOLE) GO TO 290 B2530 VPRM(IX,1Y)=FCTR1 B2540 !

CNRECH(IX,IY)=FCTR2 B2550 IF (OVERRD.NE.0) RECH(IX,1Y)=FCTR3 B256C 290 CONTINUE B2570 WRITE (6,860) ICODE,FCTR1,FCTR2 B2580 300 IF (0VERRD.NE.0) WRITE (6,1100) FCTR3 B2590 310 WRITE (6,590) B2600 00 320 IY=1,NY B2610 320 WRITE (6,840) (VPRM(IX,IY),IX=1,NX) B2621 C *************************************************************** B2630 C ---READ WATER-TABLE ELEVATION--- B2640 WRITE (6,670) B2650 READ (5,550) INPUT,FCTR B2660 00 350 IY=1,NY B2670 IF (INPUT.EQ.1) READ (5,660) (WT(IX,IY),IX=1,NX) 82680 00 340 IX=1,NX B2690 IF (INPUT.NE.1) GO TO 330 B2700 WT(IX,1Y)=WT(IX,IY)*FCTR B2710 GO TO 340 B2720 330 IF (THCK(IX,IY).NE.0.0) WT(IX,1Y)=FCTR B2730 340 CONTINUE B2740 350 WRITE (6,680) (WT(IX,1Y),IX=1,NX) B2750 C *************"************************************************** B2760 C ---SET INITIAL HEADS--- B2770 NUREG-1101, Vol. 3 C-10

00 360 IX=1,NX B2780 DO 360 IY=1,NY B2790 HI(IX,1Y)=WT(IX,IY) B2800 HC(IX,IY)=HI(IX,IY) B2810 HR(IX,IY)=HI(IX,IY) B2820 360 HK(IX,IY)=HI(IX,IY) B2830 C B2840 CALL OUTPT B2850 C *************************************************************** B2860 C ---COMPUTE ITERATION PARAMETERS--- B2870 00 370 1D=1,20 82880 A0PT(ID)=0.0 82890 370 CONTINUE B2900 ANITP=NITP-1 B2910 ALPHA 1 = 1.0 N 310 IF(HMIN.LT.2.0)ALPHAl=EXP(ALOG(1.0/HMIN)/ANITP) N 320 IF (HMIN.EQ.2.0) HMIN = 0.0 N 330 A0PT(1)=HMIN B2930 DO 380 IP=2,NITP B2940 380A0PT(IP)=A0PT(IP-1)* ALPHA 1 B2959 C B2960 WRITE (6,450) 82970 WRITE (6,460) A0PT B2980 C **************************************************************** B2990 C

---READ INITIAL CONCENTRATIONS AND COMPUTE INITIAL MASS STORED--- B3000 READ (5,550) INPUT,FCTR B3010 DO 420 IY=1,NY B3020 IF (INPUT.EQ.1) READ (5,660) (CONC (IX,IY),1X=1,NX) B3030 00 410 IX=1,NX B3040 IF (INPUT.NE.1) GO TO 390 B3050 CONC (IX,IY)= CONC (IX,IY)*FCTR B3060 GO TO 400 B3070 390 IF (THCK(IX,1Y).NE.0.0) CONC (IX,IY)=FCTR 83080 400CONINT(IX,IY)= CONC (IX,1Y) B3090 410 STORMI=STORMI+CONINT(IX,IY)*THCK(IX,IY)* AREA *POROS B3100 420 CONTINUE B3110 C **************************************************************** B3120 C ---CHECK DATA SETS FOR INTERNAL CONSISTENCY--- B3130 DO 440 IX=.* NX 83140 D0 440 IY=1,NY B3150 IF (THCK(IX,IY).GT.0.0) GO TO 430 B3160 IF (TMRX(IX,IY,1).GT.0.0) WRITE (6,940) IX,IY B3170 IF (TMRX(IX,IY,2).GT.0.0) WRITE (6,950) IX,IY B3180 i

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IF(N00EID(IX,1Y).GT.0) WRITE (6,960)IX,1Y B3190 IF (WT(IX,IY).NE.0.0) WRITE (6,970) IX,1Y B3200 IF(RECH(IX,1Y).NE.0.0) WRITE (6,980)IX,IY B3210 IF (REC (IX,1Y).NE.0.0) WRITE (6,990) IX,1Y B3220 430 IF (PERM (IX,IY).GT.0.0) GO TO 440 B3230 IF (N0DEID(IX,1Y).GT.0.0) WRITE (6,1000) IX,IY B3240 IF (WT(IX,1Y).NE.0.0) WRITE (6,10 0) IX,1Y 83250 IF(RECH(IX,IY).NE.0.0) WRITE (6,1020)IX,1Y B3260 IF (REC (IX,IY).NE.0.0) WRITE (6,1030) IX,IY B3270 IF (THCK(IX,1Y).GT.0.0) WRITE (6,1040) IX,IY B3280 440 CONTINUE B3290 C **************************************************************** B3300 RETURN B3310 C **************************************************************** B3320 C B3330 C B3340 C B3350 450 FORMAT 1H1,20HITERATION PARAMETERS) B3360 460 FORMAT 3H ,1P10E10.3/3X,1P10E10.3) B3370 470 FORMAT 1H1,27HTIME INTERVALS (IN SECONDS)) B3380 475 FORMAT (1H0,5X,65H*** WARNING *** INITIAL TIME STEP IS LONGER TH B3384 1AN PUMPING PERIOD /25X,34H*** ADJUST EITHER TINIT OR PINT.***/) B3385 480 FORMAT (1H1,15X,17HSTEADY-STATE FLOW //5X,57HTIME INTERVAL'(IN SEC) B3390 1 FOR SOLUTE-TRANSPORT SIMULATION = ,1P1E12.5) B3400 490 FORMAT (3H ,10G12.5) B3410

,20F5.1) B3420 500 FORMAT (3H 510 FORMAT (1H1,22HAQUIFER THICKNESS (FT)) B3430 530 FORMAT (IH1,30HTRANSMISSIVITY MAP (FT*FT/SEC)) B3450 540 FORMAT (20G3.0) B3460 550 FORMAT (11,G10.0) B3470 560 FORMAT (20G4.1) B3480 570 FORMAT (1H1,23HN0DE IDENTIFICATION MAP //) B3490 580 FORMAT (1H ,20I5) B3500 590 FORMAT (1H1,45HVERTICAL PERMEABILITY / THICKNESS (FT/(FT*SEC))) B3510 600 FORMAT (1HO,10X,12HX-Y SPACING:} B3520 610 FORMAT (1H ,12X,10G12.5) B3530 620 FORMAT (1H1,24HPERMEABILTY MAP (FT/SEC)) B3540 630 FORMAT (1H0,////10X 44HNO. OF FINITE-DIFFERENCE CELLS IN AQUIFER = B3550 1 ,I4//10X,28HAREA 0F AQUIFER IN MODEL = ,1 PIE 12.5,8H SQ. FT.////1 B3560 20X 47HNZCRIT (MAX. N0. OF CELLS THAT CAN BE V0ID 0F/20X,56HPARTI B3570  !

3CLES; IF EXCEEDED,' PARTICLES ARE REGENERATED) = ,I4/) B3580 640 FORMAT (20I1) B3590 B3610 660 FORMAT (20G4.0)

NUREG-1101, Vol. 3 C-12

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670 FORMAT (1H1,11HWATER TABLE) B3620 680 FORMAT (1H ,20F5.0) B3630 690 FORMAT (1HO,10X,19HAREA 0F ONE CELL = 1 PIE 12.4) B3640 700 FORMAT (212) B3650 710 FORMAT (212,2G8.2) B3660 720 FORMAT (10A8) 83670 730 FORMAT (1HO,10A8) B3680 740 FORMAT (17I4) B3690 750 FORMAT (1H1,77HU.S.G.S. METHOD-0F-CHARACTERISTICS MODEL FOR SOLUTE B3700 1 TRANSPORT IN GROUND WATER) B3710 760 FORMAT (1H0,21X 21HI N P U T D A T A) B3720 770 FORMAT (1H0,23X,16HGRID DESCRIPT0RS//13X,30HNX (NUMBER OF COLUM B3730 i

INS) = ,I4/13X 28HNY (NUMBER OF R0WS) =,16/13X,29HXDEL (X B3740 2-DISTANCE IN FEET) = ,F7.1/13X,29HYDEL (Y-DISTANCE IN FEET) = ,F7 B3750 3.1) 83760 780 FORMAT (1H0,23X,16HTIME PARAMETERS //13X,40HNTIM (MAX. N0. OF TI B3770

= ,I6/13X,40HNPMP B3780 1ME STEPS) (N0. OF PUMPING PERIODS) 2 = ,16/13X,39HPINT (PUMPING PERIOD IN YEARS) =,F11.3/13X,39 B3791 3HTIMX (TIME INCREMENT MULTIPLIER) =,F10.2/13X,39HTINIT (INIT B3800 4IAL TIME STEP IN SEC.) =,F8.0) B3810 790 FORMAT (1HO,14X,34HHYDROLOGIC AND CHEMICAL PARAMETERS //13X,1HS,7X, B3820 129H(STORAGE COEFFICIENT) =,5X,F9.6/13X,28HPOROS (EFFECTIVE B3830 2 P0ROSITV),8X,3H= ,F8.2/13X,39HBETA (CHARACTERISTIC LENGTH) B3840 3 = ,F7.1/13X,31HDLTRAT (RATIO 0F TRANSVERSE T0/21X,30HLONGITUDI B3850 4NAL DISPERSIVITY) = ,F9.2/13X,39HANFCTR (RATIO 0F T-YY TO T-XX) B3860 5 = ,F12.6) B3870 791 FORMAT (IHO,1X 26H***NON-DECAYING SPECIES ***) N 340 792 FORMAT (1HO,1X.28HSPECIES HALF LIFE (YEARS) = ,1 PIE 10.3, N 350 1 30H OR DECAY CONSTANT (1/ SECS) = ,1 PIE 10.3) N 360 793 FORMAT (1H0,1X,25H***N0N-SORBING SPECIES ***) N 370 794 FORMAT (IH0,1X,27HR0CK DENSITY (GRM/CM**3) = ,1 PIE 10.3, N 380 1 24HBULK DENSITY / POROSITY = ,1 PIE 10.3) N 390 795 FORMAT (IH0,1X 21H*** LINEAR SORBTION***, N 400 1 IX,29HDISTRIBUTION CONSTANT (KD) = ,1 PIE 10.3) N 410 796 FORMAT (1HO,1X,23H***LANGMUIR ISOTHERM ***, N 420 1 IX,16HRATE CONSTANT = ,1P1E10.3, N 430 2 IX,22HSATURATION CONSTANT = ,1P1E10.3) N 440 797 FORMAT (1HO,1X.25H***FREUNDLICH IS0 THERM ***, N 450 1 1X,16HRATE CONSTANT = ,1P1E10.3, N 460 2 IX,19HEXPONENT (ALPHA) = ,1 PIE 10.3) N 470 800 FORMAT (12G5.0) B3880 805 FORMAT (2IS,5F10.0) N 480 810 FORMAT (IH ,16Xt I2,5X,12,4X,12) B3890 ,

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NUREG-1101, Vol. 3 C-13

820 FORMAT (1H ,7X,214,3X,F9.4,3X,F8.2) B3895 830 FORMAT (1H1,39HDIFFUSE RECHARGE AND DISCHARGE (FT/SEC)) B3910 840 FORMAT (1H ,1P10E10.2) B3920 850 FORMAT (12,3G10.2,I2) B3930 860 FORMAT (1H0,7X,12,7X,1 PIE 10.3,4X,1 PIE 10.3) B3940 870 FORMAT (1H0,21X 20HEXECUTION PARAMETERS //13X,39HNITP (N0. OF ITE B3950 1 RATION PARAMETERS) = ,I4/13X,39HT0L (CONVERGENCE CRITERIA - ADI B3960 2P) = ,F9.4/13X,39HITMAX (MAX.NO.0F ITERATIONS - ADIP) = ,I4/13X,3 B3970 34HCELDIS (MAX. CELL DISTANCE PER MOVE /24X,28H0F PARTICLES - M.O.C.) B3980 4 = ,F8.3/13X,30HNPMAX (MAX. N0. OF PARTICLES),7X,2H= ,I4/12X,3 B3990 52H NPTPND (N0. PARTICLES PER N0DE),6X,3H= ,I4) B4000 880 FORMAT (1HO,5X,47H*** WARNING *** NPTPND MUST EQUAL 4,5,8, OR 9.) B4010 885 FORMAT (1HO,5X,38H*** WARNING *** NITP MUST BE POSITIVE) B4015 890 FORMAT (IH0,23X,15HPROGRAM OPTIONS //13X,30HNPNT (TIME STEP INTER B4020 1 VAL FOR/21X,18HCOMPLETE PRINT 0VT),7X,3H= ,I4/13X,31HNPNTMV (M0VE B4030 2 INTERVAL FOR CHEM./21X 28HCONCENTRATION PRINT 0UT) = ,14/13X,29HN B4040 3PNTVL (PRINT OPTION-VELOCITY /21X,24H0=N0; 1=FIRST TIME STEP;/21X,1 B4050 47H2=ALL TIME STEPS),8X,3H= ,14/13X,31HNPNTD (PRINT OPTION-DISP.C B4060 50EF./21X,24H0=N0; 1=FIRST TIME STEP;/21X,17H2=ALL TIME STEPS),8X,3 B4070 6H= ,I4/13X,32HNUM0BS (N0. OF OBSERVATION WELLS /21X,28HFOR HYDR 0GR B4080 7APH PRINT 0UT) = ,14/13X,35HNREC (N0. OF PUMPING WELLS) = ,15 B4090 8/13X,24HNC0 DES (FOR N0DE IDENT.),9X,2H= ,I5/13X,25HNPNCHV (PUNCH V B4100 9ELOCITIES),8X,2H= ,15/13X,36HNPDELC (PRINT OPT.-CONC. CHANGE) = , B4110

$I4) B4120 900 FORMAT (1H0,10X,29HLOCAlidN OF OBSERVATION WELLS //17X,3HN0.,5X,1HX B4130 1,5X,1HY/) B4140 910 FORMAT (1H0,10X.28HLOCATION OF PUMPING WELLS //11X,28HX Y RA B4150 1TE(IN CFS) CONC./) B4160 920 FORMAT (1HO,5X,37HNO. OF N03E IDENT. CODES SPECIFIED = ,I2) B4170 930 FORMAT (1H0,10X,41HTHE F0LLOWING ASSIGNMENTS HAVE BEEN MADE:/5X,51 B4180 1HC0DE N0. LEAKANCE SOURCE CONC. RECHARGE) B4190 940 FORMAT (IH .5X,61H*** WARNING *** THCK.EQ.0.0 AND TMRX(X).GT.0.0 B4200 1 AT N0DE IX =,I4,6H, IY =,14) B4210 950 FORMAT (1H .5X,61H*** WARNING *** THCK.EQ.0.0 AND TMRX(Y).GT.0.0 B4220 1 AT N0DE IX =,I4,6H, IY =,I4) B4230 960 FORMAT (1H ,5X,61H*** WARNING *** THCK.EQ.0.0 AND N0DEID.GT.0.0 B4240 1 AT N0DE IX =,14,6H, IY =,14) B4250 970 FORMAT (1H .5X,56H*** WARNING *** THCK.EQ.0.0 AND WT.NE.0.0 AT N B4260 10DE IX =,I4,6H, IY =,I4) B4270 980 FORMAT (1H .5X,58H*** WARNING *** THCK.EQ.0.0 AND RECH.NE.0.0 AT B4280 1 N0DE IX =,I4,6H, IY =,14) B4290 990 FORMAT (1H ,5X,58H*** WARNING *** THCK.EQ.0.0 AND REC.NE.0.0 AT B4300 1 N0DE IX =,14,6H, IV =,14) B4310 NUREG-1101, Vol. 3 C-14

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i 1000 FORMAT (1H 5X,61H*** WARNING *** PERM.EQ.0.0 AND N0DEID.GT.0.0 B4320 1 AT N0DE IX =,I4,6H, IY =,I4) 84330 1010 FORMAT (1H.,5X,56H*** WARNING *** PERM.EQ.0.0 AND WT.NE.0.0 AT N 84340 10DE IX =,I4,6H, IY =,14) B4350 1020 FORMAT (1H ,5X 58H*** WARNING *** PERM.EQ.0.0 AND RECH.NE.0.0 AT B4360 1 N0DE IX =,I4,6H, IY =,I4) B4370 1030 r'ORMAT (IH .5X,58H*** WARNING *** PERM.EQ.0.0 AND REC.NE.0.0 AT 84380 1 N0DE IX =,I4,6H, IY =,I4) B4390 1040 FORMAT (IH .5X,58H*** WARNING *** PERM.EQ.0.0 AND THCK.GT.0.0 AT B4400 1 N0DE IX =,14,6H, IY =,I4) B4410 1050 FORMAT (1H0,5X,45H*** WARNING *** ANFCTR WAS SPECIFIED AS 0.0/23 B4420 1X,34HDEFAULT ACTION: RESET ANFCTR = 1.0) B4430 1060 FORMAT (II) BA440 1070 FORMAT (1014,3G5.0) 84450 1080 FORMAT (1H1,5X,25HSTART PUMPING PERIOD NO. ,I2//2X,75HTHE FOLLOWIN B4460 IG TIME STEP, PUMPAGE, AND PRINT PARAMETERS HAVE BCEN REDEFINED:/) B4470 1090 FORMAT (1H0,14X,9HNTIM = ,14/15X,9HNPNT = .I4/15X,9HNITP =, B4480 114/15X,9HITMAX = ,I4/15X,9HNREC = ,I4/15X,9HNPNTMV = ,I4/15X,9H B4490 2NPNTVL = ,14/15X,9HNPNTD = ,I4/15X,9HNPDELC = ,I4/15X,9HNPNCHV = B4500 3,14/15X,9HPINT = ,F10.3/15X,9HTIMX = ,F10.3/15X,9HTINIT = ,F1 B4510 40.3/) B4520 1100 FORMAT (IH .46X,1 PIE 10.3) B4530 1110 FORMAT (1H1,5X,25HSTART PUMPING PERIOD NO. ,I2//2X, B4532 1

23HN0 PARAMETERS REDEFINED /) 84533 END B4540-SUBROUTINE ITERAT C 10 C* IBM REAL*8 THRX,VPRM,HI,HR,HC,HK,WT, REC,RECH. TIM.A0PT. TITLE C 30 COIBM REAL*8 XDEL,YDEL,S AREA,SUMT, RHO,PARAM, TEST TOL, PINT,HMIN,PYR C 40 CoIBM REAL *8DXINV,DYINV,ARINV,PORINV C 45 COIBM REAL*8 B,G,W.A.C.E,F,DR,0C,TBAR,TMK,00EF,BLH,BRK,CHK,QL,BRH C 50 LEVEL 2,N0DEID,NPCELL,NP0LD, LIMB 0,IXOBS.IYOBS,THCK, PERM,TMWL,TM0BS N 485 LEVEL 2,TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT, TITLE N 486 COMMON /PRMI/ NTIM,NPMP,NPNT,NITD,N,NX,NY,NP NREC, INT,NNX,NNY,NUM0 C 60 IBS,NM0V,IMOV,NPMAX,ITMAX,NZCRIT. PRNT,NPTPND NPNTMV,NPNTVL,NPNTD,N C 70 2PNCHV,NPDELC,ICHK C 81 COMMON /PRMC/ N0DEID(42,42),NPCELL(42,42),NP0LD(42,42), LIMB 0(500), C 85 IIX0BS(10),IYOBS(10) C 86 2,NDECAY,NSORB N 490 COMMON /HEDA/ THCK(42,42), PERM (42,42).TMWL(10,50).TMOBS(50' ..< FCTR C 110 COMMON /HEDB/ TMRX(42,42,2),VPRM(42,42),M (42,42),HR(42,42),HC(42, C 120 142),HK(42,42),WT(42,42), REC (42,42),RECH(z 12), TIM (100),A0PT(42),T C 130 2ITLE(10),XDEL,YDEL S. AREA,SUMT,RH0,PARAM, TEST.TOL, PINT,HMIN,PYR C 140 COMMON / BALM / T0TLQ,TOTLQI,TPIN,TP0UT C 146 NUREG-1101, Vol. 3 C-15

COMMON /XINV/ DXINV,0YINV,ARINV.PORINV C 160 DIMENSION W(42), B(42), G(42) C 170 C *************************************************************** C 180 K0VNT=0 C 190 PQIN = 0.0 C 192 PQ0VT = 0.0 C 193 C ---COMPUTE R0W AND COLUMN--- C 200 C ---CALL NEW ITERATION PARAMETER--- C 210 10 REMN= MOD (K0UNT,NITP) C 220 IF (REMN.EQ.0) NTH =0 C 230 NTH = NTH +1 C 240 PARAM=A0PT(NTH) C 250 C *************************************************************** C 260 C ---R0W COMPUTATIONS--- C 270 TEST =0.0 C 280 RH0=S/ TIM (N) C 290 BRK=-RHO C 300 DO 50 IY=1,NY C 310 00 20 M=1,NX C 320 W(M)=0.0 C 330 B(M)=0.0 C 340 G(M)=0.0 C 350 20 CONTINUE C 360 D0 30 IX=1,NX C 370 IF (THCK(IX,IY).EQ.0.0) GO TO 30 C 380 COEF=VPRM(IX,IY) C 390 QL=-C0EF*WT(IX,IY) C 400 A=TMRX(IX-1,IY,1)*DXINV C 410 C=TMRX(IX,IY,1)*DXINV C 420 E=TMRX(IX,IY-1,2)*DYINV C 430 F=TMRX(IX,1Y,2)*DYINV C 440 TBAR=A+C+E+F C 450 TMK=TBAR*PARAM C 460 BLH=-A-C-RH0-C0EF-TMK C 470 IF (A.EQ.0.0.AND.C.EQ.0.0.AND.RH0.EQ.0.0.AND.00EF.EQ.0.0.AND. N 500 1 TMK.EQ.0.0) GO TO 30 N 510 BRH=E+F-TMK C 480 DR=BRH*HC(IX,1Y)+BRK*HK(IX,IY)-E*HC(IX,IY-1)-F*HC(IX,IY+1)+QL+RECH C 490 1(IX,1Y)+ REC (IX,IY)*ARINV C 500 C 510 W(IX)=BLH-A*B(IX-1)

B(IX)=C/W(IX) C 520 G(IX)=(DR-A*G(IX-1))/W(IX) C 530 NUREG-1101, Vol. 3 C-16

30 CONTINUE C 540 C C 550 C ---BACK SUBSTITUTION--- C 560 00 40 J=2,NX C 570 IJ=J-1 C 580 IS=NX-IJ C 590 HR(IS,1Y)=G(IS)-B(IS)*HR(IS+1,IY) N 520 40 IF (W(IS).EQ.0.0) HR(IS,IY) = HC(IS,IY) N 530 50 CONTINUE C 610 C *************************************************************** C 620 C ---COLUMN COMPUTATIONS--- C 630 DO 90 IX=1,NX C 640

-DO 60 M=1,NY C 650 W(M)=0.0 C 660 B(M)=0.0 C 670 60 G(M)=0.0 C 680 D0 70 IY=1,NY C 690 IF(THCK(IX,IY).E C 700 COEF=VPRM(IX,IY) Q.0.0) GO T0 70 C 710 QL=-C0EF*WT(IX,IY) C 720 A=TMRX(IX,IY-1,2)*DYINV C 730 C=TMRX(IX,IY,2)*DYINV C 740 E=TMRX(IX-1,1Y,1)*DXINV C 750 F=TMRX(IX,IY,1)*DXINV C 760 TBAR=A+C+E+F C 770 TMK=TBAR*PARAM C 780 BLH=-A-C-RH0-C0EF-TMK C 790 IF(A.EQ.0.0.AND.C.EQ.0.0.AND.RH0.EQ.0.0.AND.COEF.EQ.0.0.AND. N 540 1 TMK.EQ.0.0) GO T0 70 N 550 BRH=E+F-TMK C 800 DC=BRH*HR(IX,IY)+BRK*HK(IX,IY)-E*HR(IX-1,IY)-F*HR(IX+1,IY)+QL+RECH C 810 1(IX,IY)+ REC (IX,IY)*ARINV C 820 W(IY)=BLH-A*B(IY-1) C 830 r

B(IY)=C/W(IY) C 840 l G(IY)=(DC-A*G(IY-1))/W(IY) C 850 70 CONTINUE C 860 C

C 870 C ---BACK SUBSTITL' TION--- C 880 00 80 J=2,NY C 890 IJ=J-1 C 900 IB=NY-IJ C 910 HC(IX,IB)=G(IB)-B(IB)*HC(IX,IB+1) C 920 IF (W(IB).EQ.0.0) HC(IX,IB) = HR(IX,IB) N 560 NUREG-1101, Vol. 3 C-17

IF (THCK(IX,IB).EQ.0.0) GO TO 80 C 930 CHK= ABS (HC(IX,IB)-HR(IX,IB)) C 940 IF (CHK.GT.TOL) TEST =1.0 C 950 80 CONTINUE C 960 90 CONTINUE C 970 C *************************************************************** .C 980 K0VNT=K0VNT+1 C 990 IF (TEST.EQ.0.0) GO TO 120 C1000 IF (K0VNT.GE.ITMAX) GO TO 100 C1010 GO TO 10 C1020 C **************************************************************** C1030 C ---TERMINATE PROGRAM -- ITMAX EXCEEDED--- C1040 100 WRITE (6,160) C1050 D0 110 IX=1,NX C1060 DO 110 IY=1,NY C1070 110HK(IX,IY)=HC(IX,IY) C1080 CALL OUTPT C1090 STOP C1100 C **************************************************************** C1110 C ---SET NEW HEAD (HK)--- C1120 120 D0 130 IY=1,NY C1130 D0 130 IX=1,NX C1140 IF (THCK(IX,IY).EQ.0.0) GO TO 130 C1150 HR(IX,IY)=HK(IX,IY) C1160 HK(IX,IY)=HC(IX,1Y) C1170 C C1180 C ---CUMULATIVE PUMPAGE AND RECHARGE FOR MASS BALANCE--- C1181 IF (REC (IX,IY).GT.0.0) GO TO 32 C1182 PQIN = PQIN + REC (IX,IY) C1183 GO TO 34 C1184 32 PQ0VT = PQ0VT + REC (IX,IY) C1185 34 IF (RECH(IX,IY).GT.O.0) GO TO 36 C1186 PQIN = PQIN + RECH(IX,IY)* AREA C1187 GO TO 38 C1188 36 PQ00T = PQ0VT + RECH(IX,IY)* AREA C1189 C ---COMPUTE LEAKAGE FOR MASS BALANCE--- C1190 38 IF (VPRM(IX,IY).EQ.0.0) GO TO 130 C1201 DELQ=VPRM(IX,IY)* AREA *(WT(IX,IY)-HK(IX,IY)) C1210 l IF (GELQ.GT.0.0) GO TO 125 C1215 T0TLQ=TOTLQ+DELQ* TIM (N) C1220 GO TO 130 C1222 125 T0TLQI=T0TLQI+0ELQ* TIM (N) C1224 130 CONTINUE C1230 NUREG-1101, Vol. 3 C-18

TPIN = TPIN + PQIN* TIM (N) C1232 TPOUT = TPOUT + PQ0VT* TIM (N) C1233 C C1240 WRITE (6,140) N C1250 WRITE (6,150) K0VNT C1260 C **************************************************************** C1270 RETURN C1280 C **************************************************************** C1290 C C1300 C

C1310 C

C1320 140 FORMAT (1H0//3X,4HN = .114) C1330 150 FORMAT (1H ,2X,23HNUMBER OF ITERATIONS = ,1I4) C1340 160 FORMAT (1H0,5X,64H*** EXECUTION TERMINATED -- MAX. NO. ITERATION C1350 IS EXCEEDED ***/26X,21HFINAL OUTPUT FOLLOWS:) C1360 END i

C1370-l SUBROUTINE GENPT D 10

, C* IBM REAL*8 TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT, TITLE D 20 C* IBM REAL*8 XDEL,YDEL,S, AREA SUMT,RH0,PARAM, TEST,TOL, PINT,HMIN,PYR D 30 INTEGER PTID D 35 LEVEL 2,N0DEID,NPCELL,NP0LD, LIMB 0,IX0BS,IYOBS,THCK, PERM,TMWL,TMOBS N 562 LEVEL 2,TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT, TITLE,PART, CONC N 563 LEVEL 2,TMCN,VX,VY,CONINT,CNRECH,PTID N 564 COMMON /PRMI/ NTIM,HFMP,NPNT,NITP,N,NX,NY,NP,NREC, INT,NNX,NNY,NUM0 D 40 IBS,NMOV,IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N D 50 2PNCHV,NPDELC,ICHK D 61 COMMON /PRMC/ N0DEID(42,42),NPCELL(42,42),NP0LD(42,42), LIMB 0(500), D 65 IIX0BS(10),IY0BS(10) D 66 2,NDECAY,NSORB N 570 COMMON /HEDA/THCK(42,42), PERM (42,42),TMWL(10,50).TM0BS(50),ANFCTR D 90 COMMON /HEDB/ TMRX(42,42,2),VPRM(42,42),HI(42,42),HR(42,42),HC(42, D 100 142),HK(42,42),WT(42,42), REC (42,42),RECH(42,42), TIM (100),A0PT(42),T D 110 21TLE(10),XDEL,YDEL,S, AREA SUMT, RHO,PARAM, TEST,TOL, PINT,HMIN,PYR D 120 COMMON /CHMA/ PART(3,9000), CONC (42,42) TMCN(10,50),VX(42,42),VY(42 0 130 1,42),CONINT(42,42),CNRECH(42,42),POR0S.SUMTCH, BETA,TIMV, STORM.STOR D 140 2MI CMSIN, CMS 0VT,FLMIN,FLMOT,50MIO,CELDIS,DLTRAT,CSTORM D 150 3,DCYLAM,BLKDEN,SRBRAT,SRBSAT,SRBALF,VOLDCY,VOLSRB,SRBDCY N 580 COMMON /CHMP/ PTID(9000) D 155 DIMENSION RP(8), RN(8), IPT(8) D 160 C *************************************************************** D 170 Fl=0.30 D 180 F2=1.0/3.0 D 190 IF (NPTPND.EQ.4) Fl=0.25 D 200 NUREG-1101, Vol. 3 C-19

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i D 210

( IF (NPTPND.EQ.9) Fl=1.0/3.0 IF (NPTPND.EQ.8) F2=0.25 0 220

( D 230 NCHK=NPTPND l IF (NPTPND.EQ.5.0R.NPTPND.EQ.9) NCHK=NPTPND-1 D 240 IF (TEST.GT.98.) GO TO 10 0 250 C *************************************************************** D 260 C ---INITIALIZE VALUES--- D 270 STORM =0.0 0 280 CMSIN=0.0 D 290 CMSOUT=0.0 D 300 FLMIN=0.0 0 310 FLM0T=0.0 D 320 SUMIO=0.0 D 330 C *************************************************************** D 340 10 00 20 IN=1,NPMAX D 345 PTID(IN)=0 D 355 D0 20 ID=1,3 0 365 20PART(ID.IN)=0.0 D 370 00 30 IA=1,8 0 380 RP(IA)=0.0 0 390 RN(IA)=0.0 D 400 30IPT(IA)=0 0 410 C ---SET UP LIMB 0 ARRAY--- D 420 DO 40 IN=1,500 0 430 40 LIMB 0(IN)=0.0 D 440 IND=1 0 450 DO 50 IL=1,500,2 D 460 LIMB 0(IL)=IND D 470 50 IND=IND+1 D 480 C *************************************************************** D 490 C ---INSERT PARTICLES--- D 500 00 410 IX=1,NX D 510 DO 410 IY=1,NY D 520 IF (THCK(IX,IY).EQ.0.0) GO TO 410 D 530 KR=0 D 540 TEST 2=0.0 D 550 METH=1 0 560 NPCELL(IX,IY)=0 D 570 NPOLD(IX,1Y)=NPTPND D 575 C1= CONC (IX,IY) D 580 IF (C1.LE.1.0E-05) TEST 2=1.0 0 590 IF (VPRM(IX,IY).GT.0.09) TEST 2=1.0 D 600 IF (REC (IX,1Y).NE.0.0) TEST 2=1.0 D 610

(

( NUREG-1101, Vol. 3 C-20 t . .

J IF (THCK(IX+1,IY+1).EQ.0.0.0R.THCK(IX+1,IY-1).EQ.0.0.0R.THCK(IX-1 D 620 IIY+1).EQ.0.0.0R.THCK(IX-1,IY-1).EQ.0.0) TEST 2=1.0 D 630 IF((THCK(IX,1Y+1).EQ.0.0.0R.THCK(IX,IY-1).EQ.0.0.0R.THCK(IX+1,IY) D 640 1.EQ.0.0.0R.THCK(IX-1,IY).EQ.0.0).AND.NPTPND.GT.5) TEST 2=1.0 D 650 CN0DE Cl*(1.0-F1) D 660 IF (TEST.LT.98.0.0R. TEST 2.GT.0.0) GO TO 70 D 670 i SUMC= CONC (IX+1,IY)+ CONC (IX-1,IY)+ CONC (IX,IY+1)+ CONC (IX,IY-1) D 680

'IF (NCHK.EQ.4) GO TO 60 0 690 SUMC=SUMC+ CONC (IX+1,IY+1)+ CONC (IX+1,IY-1)+ CONC (IX-1,IY+1)+ CONC (IX- D 700 11,IY-1) D 710 60 AVC=SUMC/NCHK D 720 IF (AVC.GT.C1) METH=2 0 730 C

D 740 C ---PUT 4 PARTICLES ON CELL DIAGONALS--- D 750 70 D0 140 IT=1,2 D 760 EVET=(-1.0)**IT D-770 00 140 IS=1,2 D 780 EVES =(-1.0 **IS 0 790 PART(1,IND=IX+Fl*EVET D 800 PART(2,IND =IY+Fl* EVES

' 3 810 PART(2,IND=-PART(2,IND) D 820 PART(3,IND =C1

~

D 830 KR=KR+1 0 832 PTID(IND)=KR D 834 IF (TEST.LT.98.0.0R. TEST 2.GT.0.0) GO TO 130 0 840 IXD=IX+EVET D 850 IYD=IY+ EVES D 860 IPT(KR)=IND D 880 IF (METH.EQ.2) GO TO 80 D 890 j

PART(3,IND)=CN0DE+ CONC (IXD,1YD)*F1 D 900 GO TO 90 D 910 l 80 PART(3,IND)=2.0*Cl* CONC (IXD,IYD)/(C1+ CONC (IXD,IYD)) D 920 t i

90IF(Cl-CONC (IXD,IYD)) 100,110,120 D 930 100RP(KR)= CONC (IXD,IYD)-PART(3,IND) D 940 RN(KR)=Cl-PART(3,IND)

D 950 GO TO 130 D 960 l

110 RP(KR)=0.0 0 970 RN(KR)=0.0 D 980 GO TO 130 D 990 120 RP(KR)=Cl-PART(3,IND) 01000 RN(KR)= CONC (IXD,IYD)-PART(3,IND) 130 IND=IND+1 01010 140 CONTINUE D1020 C D1030 D1040 NUREG-1101, Vol. 3 C-21

IF (NPTPND'.EQ.5.0R.NPTPND.EQ.9) GO TO 150 01050 GO TO 160 D1060 C ---PUT ONE PARTICLE AT CENTER OF CELL--- D1070 150PART(1,IND=IX D1075 PART(2,IND =-IY D1090 PART(3,1ND =C1 01100 D1105 PTID(IND)=5 IND=IND+1 D1110 C ---PLACE NORTH, SOUTH, EAST, AND WEST PARTICLES--- 01120 D1130 160 IF (NPTPND.LT.8) GO TO 290 CN0DE=Cl*(1.0-F2) 01140 DO 280 IT=1,2 D1150

-EVET=(-1.0)**IT D1160 PART(1,IND)=IX+F2*EVET D1170 PART(2,1ND)=-IY D1180 PART(3,IND)=C1 D1190 01192 IF(EVET.LT.0)PTID(IND)=6 D1194 IF (EVET.GT.0) PTID(IND)=8 01200 IF (TEST.LT.98.0.0R. TEST 2.GT.0.0) GO TO 220 IXD=IX+EVET D1210 KR=KR+1 01220 D1230 IPT(KR)=IND 01240 IF (METH.EQ.2) GO TO 170 PART(3,IND)=CN0DE+ CONC (IXD,1Y)*F2 D1250 GO TO 180 D1260 170 PART(3,IND)=2.0*Cl* CONC (IXD,1Y)/(C1+ CONC (IXD IY)) D1270 180 IF (Cl-CONC (IXD,IY)) 190,200,210 D1280 190 RP(KR)= CONC (IXD,1Y)-PART(3,IND) D1290 RN(KR)=Cl-PART(3,IND) 01300 GO TO 220 D1310 D1320 200 RP(KR)=0.0 D1330

-RN(KR)=0.0 GO TO 220 01340 210 RP(KR)=Cl-PART(3,IND) D1350 RN(KR)= CONC (IXD,IY)-PART(3,1ND) 01360 PART(1,IND)=IX D1380 220 IND=IND+1 01370 PART(2,1ND)=IY+F2*EVET D1390 l PART(2,IND)=-PART(2,IND) D1400 PART(3,1ND)=C1 D1410.

D1412 IF (EVET.LT.0) PTID(IND)=7 D1414 IF (EVET.GT.0) PTID(IND)=9 D1420 IF (TEST.LT.98.0.0R. TEST 2.GT.0.0) GO TO 280 NUREG-1101, Vol. 3 C-22

IYD=IY+EVET D1430 KR=KR+1 01440 IPT(KR)=IND D1450 IF (METH.EQ.2) GO TO 230 D1460 PART(3,IND)=CN0DE+ CONC (IX,IYD)*F2 D1470 l GO TO 240 01480 l 230 PART(3,1ND)=2.0*Cl* CONC (IX,1YD)/(C1+ CONC (IX,IYD)) D1490 l 240IF(Cl-CONC (IX,IYD)) 250,260,270 D1500 250 RP(KR)= CONC (IX,IYD)-PART(3,IND) D1510 RN(KR)=Cl-PART(3,IND) D1520 GO TO 280 D1530 260RP(KR)=0.0 D1540 RN(KR)=0.0 D1550 GO TO 280 D1560 270 RP(KR)=Cl-PART(3,IND) DI570 RN(KR)= CONC (IX,1YD)-PART(3,IND) D1580 280 IND=IND+1 D1590 C 01600 290 IF (TEST.LT.98.0.0R. TEST 2.GT.0.0) GO TO 410 D1610

-SUMPT=0.0 D1620 C ---COMPUTE CONC. GRADIENT WITHIN CELL--- D1630 DO 300 KPT=1,NCHK D1640 IK=IPT(KPT) D1650 300 SUMPT=PART(3,IK)+SUMPT 01660 CBAR=SUMPT/NCHK D1670 C ---CHECK MASS BALANCE WITHIN CELL AND ADJUST PT. CONCS.--- D1680 SUMPT=0.0 01690 IF (CBAR-C1) 310,410,330 D1700 310 CRCT=1.0-(CBAR/C1) D1710 IF (METH.EQ.1) CRCT=CBAR/C1 D1720 00 320 KPT=1,NCHK D1730 IK=IPT(KPT) D1740 PART(3,IK)=PART(3,IK)+RP(KPT)*CRCT D1750 320 SUMPT=SUMPT+PART(3,IK) 01760 CBARN=SUMPT/NCHK D1770 GO TO 350 D1780 330 CRCT=1.0-(C1/CBAR) D1790 IF (METH.EQ.1) CRCT=C1/CBAR D1800 00 340 KPT=1,NCHK D1810 IK=IPT(KPT) 01820 PART(3,IK)=PART(3,IK)+RN(KPT)*CRCT D1830 340 SUMPT=SUMPT+PART(3,IK) 01840 CBARN=SUMPT/NCHK D1850 NUREG-1101, Vol. 3 C-23

350 IF (CBARN.EQ.C1) GO TO 410 D1860 C ---CORRECT FOR OVERCOMPENSATION--- D1870 CRCT=C1/CBARN D1880 DO 380 KPT=1,NCHK D1890 IK=IPT(KPT) D1900 PART(3,1K)=PART(3,IK)*CRCT D1910 C ---CHECK CONSTRAINTS--- D1920 IF (PART(3,IK)-C1) 360,380,370 D1930 360 CLIM=Cl-RP(KPT)+RN(KPT) D1940 IF (PART(3,IK).LT.CLIM) GO TO 390 D1950 GO TO 360 D1960 370 CLIM=C1+RP(KPT)-RN(KPT) D1970 IF (PART(3,1K).GT.CLIM) GO TO 390 D1980 380 CONTINUE D1990 GO TO 410 D2000 390 TEST 2=1.0 D2010 DO 400 KPT=1,NCHK -

D2020 IK=IPT(KPT) D2030 400 PART(3,1K)=C1 02040 410 CONTINUE D2050 NP=IND 02060 IF (INT.EQ.0) CALL CHMOT D2070 C **************************************************************** D2080 RETURN 02090 C **************************************************************** D2100 END D2110-SUBROUTINE VELO E 10 C* IBM REAL*8 TMRX,VPRM,HI,HR,HC.HK,WT, REC,RECH, TIM,A0PT, TITLE E 30 C* IBM REAL *8DXINV,DYINV,ARINV,PORINV E 35 C* IBM REAL*8 XDEL,YDEL,S, AREA,SUMT,RH0.PARAM, TEST,TOL, PINT,HMIN,PYR E 40 C* IBM REAL*8 RATE,SLEAK,DIV . E 50 LEVEL 2,N0DEID,NPCELL,NPOLD, LIMBO,IX0BS.IY0BS,THCK, PERM,TMWL,TMOBS N 582 LEVEL 2,TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT TITLE,PART, CONC N 583 LEVEL 2,TMCN,VX,VY,CONINT,CNRECH,SUMC,VXBDY,VYBDY, DISP N 584 LEVEL 2,VXNORD,VYNORD N 585 COMMON /PRMI/ NTIM,NPMP,NPNT,NITP,N,NX,NY,NP,NREC, INT,NNX,NNY,NUM0 E 60 IBS NMOV,IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N E 70 2PNCHV,NPDELC,1CHK E 81 COMMON /PRMC/ N0DEID(42,42),NPCELL(42,42),NPOLD(42,42), LIMB 0(500), E 85 IIX0BS(10),IY0BS(10) E 86 2,NDECAY,NSORB N 590 COMMON /HEDA/ THCK(42,42), PERM (42,42),TMWL(10,50),TM0BS(50),ANFCTR E 110 COMMON /HEDB/ TMRX(42,42,2),VPRM(42,42),HI(42,42),HR(42,42),HC(42, E 120 NUREG-1101, Vol. 3 C-24

142),HK(42,42),WT(42,42), REC (42,42),RECH(42,42) TIM (100),A0PT(42),T E 130 21TLE(10),XDEL,YDEL,S AREA,SUMT,RH0,PARAM, TEST,TOL, PINT,HMIN,PYR E 140 COMMON /XINV/ DXINV,DYINV,ARINV,PORINV E 150 COMMON /CHMA/ PART(3,9000), CONC (42,42),TMCN(10,50),VX(42,42),VY(42 E 160 1,42),CONINT(42,42),CNRECH(42,42),POROS,5UMTCH, BETA.TIMV, STORM.STOR E 170 2MI,CMSIN, CMS 0VT,FLMIN,FLMOT,SUMIO,CELDIS,DLTRAT,CSTORM E 180 3,DCYLAM,BLKDEN,SRBRAT,SRBSAT,SRBALF,V0LDCY,VOLSRB.SRBDCY N 600 COMMON /VNORD/ VXNORD(42,42),VYNORD(42,42) N 605 COMMON /CHMC/ SUMC(42,42),VXBDY(42,42),VYBDY(42,42) E 190 COMMON /DIFUS/ DISP (42,42,4) E 200 C *************************************************************** E 210 C ---COMPUTE VELOCITIES AND STORE--- E 220 VMAX=1.0E-10 E 230 VMAY=1.0E-10 E 240 VMXBD=1.0E-10 E 250 VMYBD=1.0E-10 E 260 TMV = 1.0E5* TIM (N) E 275 LIM =0 E 280 MAXX = 0 E 284 MAXY = 0 E 285 C

E 290 D0 20 IX=1,NX E 300 D0 20 IY=1,NY E 310 D0 10 IZ=1,4 E 320 10 DISP (IX,IY,IZ)=0.0 E 330 C

E 340 IF (THCK(IX,IY).EQ.0.0) GO TO 20 E 350 DIST = 0.0 N 610 IF (NSORB.LT.1) GO TO 6 N 620 IF (NSORB.GT.1) GO TO 3 N 630 C*** ****************** LINEAR SORBTION*************** N 640 DIST = SRBRAT*BLKDEN N 650 GO T0 6 N 660 3 IF (NSORB.GT.2) GO TO 4 N 670 C*** ******************LANGMUIR S0RBTION************* N 680 DIST = BLKDEN*SRBRAT*SRBSAT/(1.0+SRBRAT* CONC (IX,IY))**2.0 N 690 GO TO 6 N 700 4 IF (SRBALF.EQ.0.0) GO TO 6 N 710 C*** ******************FREUNDLICH S0RBTION*********** N 720 LOGCON = -23.0 N 730 IF (CONC (IX,1Y).GT.1.0E-10) LOGCON = ALOG(CONC (IX,IY)) N 740 5 SRBEXP = (SRBALF-1.0)*LOGCON N 750  ;

IF (SRBEXP.GT.23.0) SRBEXP = 23.0 N 760 NUREG-1101, Vol. 3 C-25 l

DIST = BLKDEN*SRBALF*SRBRAT*EXP(SRBEXP) N 770 6 RETARD = 1.0/(1.0+DIST) N 780 RATE = REC (IX,1Y)/ AREA E 360 SLEAK=(HK(IX,IY)-WT(IX,1Y))*VPRM(IX,1Y) E 370 DIV= RATE +SLEAK+RECH(IX,1Y) E 380 C

E 390 C ---VELOCITIES AT N0 DES--- E 400 C ---X-DIRECTION--- E 410 DHX=HK(IX-1,IY)-HK(IX+1,IY) E 421 IF THCKIX-1,1Y).EQ.0.0)DHX=HK(IX,1Y)-HK(IX+1,IY) E 431 IF THCK IX+1,IY).EQ.0.0) DHX=HK(IX-1,1Y)-HK(IX,IY) E 441 IF THCK IX-1,IY).EQ.0.0.AND.THCK(IX+1,1Y).EQ.0.0) DHX=0.0 E 451 GRDX=DHX*DXINV*0.50 E 455 VX(IX,IY)= PERM (IX,1Y)*GRDX*PORINV E 460 VXNORD(IX,IY)=VX(IX,1Y) N 791 VX(IX,IY)=VX(IX,IY)* RETARD N 795 ABVX= ABS (VX(IX,1Y)) E 470 E 480 IF (ABVX.GT.VMAX) VMAX=ABVX C ---Y-DIRECTION--- E 490 DHY=HK(IX,1Y-1)-HK(IX,IY+1) E 501 IF(THCK(IX,IY-1.EQ.0.0)DHY=HK(IX,1Y)-HK(IX,IY+1) E 511 IF (THCK(IX,1Y+1.EQ.0.0) DHY=HK(IX,IY-1)-HK(IX,1Y) E 521 IF (THCK(IX,IY-1 .EQ.0.0.AND.THCK(IX,1Y+1).EQ.0.0) DHY=0.0 E 531 GRDY=DHY*DYINV*0.50 E 535 VY(IX,IY)= PERM (IX,IY)*GRDY*PORINV*ANFCTR E 540 VYNORD(IX,IY)=VY(IX,IY) N 801 VY(IX,IY)=VY(IX,IY)* RETARD N 805 ABVY= ABS (VY(IX,IY)) E 550 E 560 IF(ABVY.GT.VMAY)VMAY=ABVY E 570 C

C ---VELOCITIES AT CELL BOUNDARIES--- E 580 GRDX=(HK(IX,IY)-HK(IX+1,IY))*DXINV E 590 PERMX=2.0* PERM (IX,IY)* PERM (IX+1,IY)/(PERM (IX,1Y)+ PERM (IX+1,IY)) E 600 VXBDY(IX,1Y)=PERMX*GRDX*PORINV E 610

• RETARD N 810 1

GRDY=(HK(IX,1Y)-HK(IX,1Y+1))*DYINV E 620 PERMY=2.0* PERM (IX,IY)* PERM (IX,IY+1)/(PERM (IX,1Y)+ PERM (IX,IY+1)) E 630 VYBDY(IX,1Y)=PERMY*GRDY*PORINV*ANFCTR E 640

• RETARD N 820 1

ABVX= ABS (VXBDY(IX,IY)) E 650 ABVY= ABS (VYBDY(IX,IY)) E 660 E 670 IF{ABVX.GT.VMXBD)VMXBD=ABVX E 680 IF(ABVY.GT.VMYBD)VMYBD=ABVY E 690 C

NUREG-1101, Vol. 3 C-26

l IF (DIV.GE.0.0) GO TO 20 E 700 TDIY=(POROS*THCK(IX,IY))/ ABS (DIV) E 710 IF (TDIV.GE.TMV) GO TO 20 E 722 TMV = TDIV E 724 MAXX = IX E 725 MAXY = IY E 726 20 CONTINUE E 730 C *************************************************************** E 740 C ---PRINT VELOCITIES--- E 750 IF (NPNTVL.EQ.0) GO TO 80 E 760 IF (NPNTVL.EQ.2) GO TO 30 E 770 IF (NPNTVL.EQ.1.AND.N.EQ.1) GO TO 30 E 780 GO TO 80 E 790 30 WRITE (6,320) E 800 WRITE (6,330) E 810 00 40 IY=1,NY E 820 40 WRITE (6,350) (VX(IX,IY),IX=1,NX) E 830 WRITE (6,340) E 840 DO 50 IY=1,NY E 850 50 WRITE (6,350) (VXBDY(IX,IY),IX=1,NX) E 860 WRITE (6.360) E 870 WRITE (6,330) E 880 DO 60 IY=1,NY E 890 60 WRITE (6,350)(VY(IX,IY),IX=1,NX) E 900 WRITE (6,340) E 910 DO 70 IY=1,NY E 920 70 WRITE (6,350) (VYBDY(IX,IY),IX=1,NX) E 930 C ---PRINT UNRETARDED X AND Y VELOCITIES--- N 822A IF(NSORB.LT.1)GOTO80 N 822B WRITE (6,362) N 822C l WRITE (6,330) N 8220

! 00 72 IY=1,NY N 822E 72 WRITE (6,350)(VXN0PD(IX,IY),1X=1,NX) N 822F WRITE (6,364) N 822G WRITE (6,330) N 822H DO 74 IY=1,NY N 822I 74 WRITE (6,350) (VYNCRD(IX,IY),IX=1,NX) N 822J C ---PUNCH VELOCITIES--- E 940 80 IF (NPNCHV.EQ.0) GO TO 110 E 950 IF (NPNCHV.EQ.2) GO TO 90 E 960 IF (NPNCHV.EQ.1.AND.N.EQ.1) GO TO 90 E 970 GO TO 110 E 980 90 WRITE (7,510) NX,NY,XDEL,YDEL,VVAX,vPAY E 990 NUREG-1101, Vol. 3 C-27

D0 100 IY=1,NY E1000 WRITE (7,520)(VX(IX,IY),1X=1,NX) E1010 100 WRITE (7,520) (VY(IX,IY) IX=1,NX) E1020 C **************************************************************** E1030 C ---COMPUTE NEXT TIME STEP--- E1040 110 WRITE (6,390) E1050 WRITE (6,400)VMAX,VMAY E1060 WRITE (6.410) VMXBD,VMYBD E1070 TDELX=CELDIS*XDEL/VMAX E1080 TDELY=CELDIS*YDEL/VMAY E1090 TDELXB=CELDIS*XDEL/VMXBD - E1100 TDELYB=CELDIS*YDEL/VMYBD E1110 TIMV= AMIN 1(TDELX,TDELY,TDELXB.TDELYB) E1120 IF(AMAX1(VMAX,VMAY,VMXBD,VMYBD).LE.1.0E-10) WRITE (6,570) E1125 WRITE (6.310) TMV,TIMV E1130 IF (TMV.LT.TIMV) GO TO 120 E1140 LIM =-1 E1150 GO TO 130 E1160 120 TIMV=TMV E1170 LIM =1 E1180 130 NTIMV= TIM (N)/TIMV E1190 NMOV=NTIMV+1 E1200 WRITE (6,420) TIMV,NTIMV,NM0V E1210 TIMV= TIM (N)/NMOV E1220 WRITE (6,370) TIM (N) E1230 WRITE (6.380) TIMV E1240 C E1250 IF (BETA.EQ.0.0) GO TO 200 E1260 C **************************************************************** E1270 C ---COMPUTE DISPERSION COEFFICIENTS--- E1280 ALPHA = BETA E1290 ALNG= ALPHA E1300 TRAN=DLTRAT* ALPHA E1310 XX2=XDEL*XDEL E1320 YY2=YDEL*YDEL E1330 XY2=4.0*XDEL*YDEL E1340 D0 150 IX=2,NNX E1350 D0 150 IY=2,NNY E1360 IF (THCK(IX,IY).EQ.0.0) GO TO 150 E1370 VXE=VXBDY(IX,1Y) E1380 VYS=VYBDY(IX,IY) E1390 IF (THCK(IX+1,1Y).EQ.0.0) GO TO 140 E1400 C ---FORWARD COEFFICIENTS: X-DIRECTION--- E1410 l NUREG-1101, Vol. 3 C-28 l

l

VYE=(VYBDY(IX,IY-1)+VYBDY(IX+1,IY-1)+VYS+VYBDY(IX+1,I())/4.0 E1420 VXE2=VXE*VXE E1430 VYE2=VYE*VYE E1440 VMGE=SQRT(VXE2+VYE2) E1450 IF (VMGE.LT.1.0E-20) GO TO 140 E1460 DALN=ALNG*VMGE E1470 DTRN=TRAN*VMGE E1480 VMGE2=VMGE*VMGE E1490 C ---XX COEFFICIENT--- E1500 DISP (IX,1Y,1)=(DALN*VXE2+DTRN* BYE 2)/(VMGE2*XX2) E1510 C ---XY COEFFICIENT--- E1520 DISP (IX,1Y,3)=(DALN-DTRN)*VXE*VYE/(VMGE2*XY2) E1530 C ---FORWARD C0EFFICIENTS: Y-DIRECTION--- E1540 140 IF (THCK(IX,1Y+1).EQ.0.0) GO TO 150 E1550 VXS=(VXBDY(IX-1,IY)+VXE+VXBDY(IX-1,IY+1)+VXBDY(IX,IY+1))/4.0 E1560 VYS2=VYS*VYS E1570 VXS2=VXS*VXS E1580 VMGS=SQRT(VXS2+VYS2) E1590 IF (VMGS.LT.1.0E-20) GO TO 150 E1600 DALN=ALNG*VMGS E1610 DTRN=TRAN*VMGS E1620 VMGS2=VMGS*VMGS E1630 C ---YY COEFFICIENT--- '

E1640 DISP (IX,1Y,2)=(DALN*VYS2+DTRN*VXS2)/(VMGS2*YY2) E1650 C ---YX COEFFICIENT--- E1660 DISP (IX,1Y,4)=(DALN-DTRN)*VXS*VYS/(VMGS2*XY2) E1670 150 CONTINUE E1680 C **************************************************************** E1690 C

---ADJUST CROSS-PRODUCT TERMS FOR ZERO THICKNESS--- E1700 00 160 IX=2,NNX E1710 00 160 IY=2,NNY E1720 IF(THCK(IX,IY+1).EQ.0.0.0R.THCK(IX+1,IY+1).EQ.0.0.0R.THCK(IX,IY-1 E1730 1).EQ.0.0.0R.THCK(IX+1,IY-1).EQ.0.0) DISP (1X,IY,3)=0.0 E1740 IF(THCK(IX+1,IY).EQ.0.0.0R.THCK(IX+1,IY+1).EQ.0.0.0R.THCK(IX-1,IY E1750 1).EQ.0.0.0R.THCK(IX-1,1Y+1).EQ.0.0) DISP (IX,1Y,4)=0.0 E1760 160 CONTINUE E1770 C **************************************************************** E1780 C

---CHECK FOR STABILITY OF EXPLICIT METHOD--- E1790 TIMDIS=0.0 D0 170 IX=2,NNX E1800 00 170 IY=2,NNY E1810 E1820 TDC0= DISP (IX,IY,1)+ DISP (IX,IY,2)

E1830 170 IF (TDCO.GT.TIMDIS) TIMDIS=TDC0 E1840 NUREG-1101, Vol. 3 C-29

TIMDC=0.5/TIMDIS E1850 WRITE (6.440) TIMDC E1860 NTIMD= TIM (N)/TIMDC E1870 NDISP=NTIMD+1 E1880 E1890 IF (NDISP.LE.NMOV) GO TO 180 NMOV=NDISP E1900 TIMV= TIM (N)/NM0V E1910 LIM =0 E1920 180 WRITE (6,430) TIMV,NTIMD,NMOV E1930 C **************************************************************** E1940 C ---ADJUST DISP. EQUATION C0EFFICIENTS FOR SATURATED THICKNESS--- E1950 D0 190 IX=2,NNX E1960 D0 190 IY=2,NNY E1970 IF (THCK(IX,IY).EQ.0.0) GO TO 190 E1972 BAVX=2.0*THCK(IX,IY)*THCK(;X+1,IY)/(THCK(IX,1Y)+THCK(IX+1,IY)) E1974 BAVY=2.0*THCK(IX,IY)*THCK(IX,IY+1)/(THCK(IX,IY)+THCK(IX,IY+1)) E1976 DISP (IXIY,1)= DISP (IX,IY,1)*BAVX E2000 DISP (IX,IY,2)= DISP (IX,1Y,2)*BAVY E2010 DISP (IX,IY,3)= DISP (IX,IY,3)*BAVX E2020 DISP (IX,IY,4)= DISP (IX,1Y,4)*BAVY E2032 E2034 190 CONTINUE C **************************************************************** E2040 E2052 200 IF (NMOV.LT.2) GO TO 235 IF (LIM) 210,220,230 E2054 210 WRITE (6,536) E2060 GO TO 240 E2070 220 WRITE (6,540) E2080 GO TO 240 E2090 230 WRITE (6,550) E2100 WRITE (6,560) MAXX,MAXY E2102 E2104 GO TO 240 235 WRITE (6,580) E2106 C **************************************************************** E2110 E2120 C ---PRINT DISPERSION EQUATION COEFFICIENTS---

E2130 240 IF (NPNTD.EQ.0) GO TO 300 E2140 IF (NPNTD.EQ.2) GO TO 250 E2150 IF (NPNTD.EQ.1.AND.N.EQ.1) GO TO 250 E2160 GO TO 300 l 250 WRITE (6,450) E2170 WRITE (6,460) E2180 D0 260 IY=1,NY E2190 260 WRITE (6,500) (DISP (IX,IY,1),IX=1,NX) E2200 WRITE (6,470) E2210 NUREG-1101, Vol. 3 C-30

y M - ..

ojl ,

.k a -

e

~ '

r <

00 270 IY=1,NY ,

E2220 270 WRITE (6,500)(DISP (IX,IY,2)',1X=1,NX) E2230 WRITE (6,480) E2240 DO 280 IY=1,NY .

E2250 280 WRITE (6,500)(DISP (IX,IY,3),IX=1,NX) E2260 WRITE'(6,490) E2270 D0 290 IY=1,NY - E2280 290 WRITE (6,500) (DISP (IX,1Y,4),IX=1,NX) E2290 C ****************************************************************

' E2300 300 RETURN ,

E2310 C *********d****************************************************** E2320 C b

• < E2330 C E2340 C

E2350 310 FORMAT (1H ,19H TMV (MAX.+INJ.) = ,G12.5(20H TIMV (CELDIS) =, E2360 11P1E12.5) i E2370

'320 FORMAT (1H1,12HX VELOCITIES). e E2380 t

330 F0FMAT (1H ,25X,8 HAT N0 DES /) E2390 340 FORMAT (1H0,25413 HON BOUNDARIES /) '-

E2400 350 FORMAT (1H ,1P10E12.3) i E2410 360 FORMAT (IH1,12HY VELOCITIES) E2420 362 FORMAT (IH1,24H UNRETARDED X VELOCITIES) N 823 364' FORMAT (1H1,24H UNRETARDED Y VELOCITIES) N 824 370 FORMAT (3H 11HTIM (N) =.1 PIE 12.5) .

E2430 380 FORMAT (3H ,11RTIMEVELO = ,1 PIE 12.5) E2440 390 FORMAT (IH1,10X,29HSTABILITY CRITERIA --- M.0.C.//) E2450 400 FORMAT (IH0,8H VMAX d ,1 PIE 9.2,5X,7HVMAY = .1 PIE 9.2) E2460 410 FORMAT (IH ',8H VMX8D= ,1P1E9.2,5X,7HVMYBD= ,1 PIE 9.2) E2470 420 FORMAT (IHO,8H TIMV = ,1P1E9.2,5X,8HNTIMV =',I5,5X,7HNM0V = I5/) E2480 430 FORMAT (IHO,8H TIMV = ,1P1E9.2,5X,8HNTIMO = ,IS,5X,7HNMOV = ,IS) E2490 440 FORMAT (3H 11HTIMEDISP = .1 PIE 12.5) E2500 i

450 FORMAT.(1H1,32HDISPERS10N EQUATION C0EFFICIENTS,10X,25H=(D-IJ)*(B) E2510 1/(GRID FACTOR))~ E2520 460 FORMAT (1H 35X,14HXX COEFFICIENT /) E2530 470 FORMAT (1H 35X,14HYY C0 EFFICIENT /) E2540 480 FORMAT (1H .35X,14HXY COEFFICIENT /) E2550 l

490 FORMAT (1H ,35X,14HYX COEFFICIENT /) E2560 500 FORMAT (1H 1P10E8.1) E2570 510 FORMAT (2I4,2F10.1,2F10.7) E2580 520 FORMAT (8F10.7) E2590 i 530 FORMAT (1H0,10X,42HTHE LIMITING STABILITY CRITERION IS CELDIS) E2600 540 FORMAT (1H0,10X,40HTHE LIMITING STABILITY CRITERION IS BETA) E2610 550 FORMAT (IHO,10X,58HTHE LIMITIN3 STABILITY CRITERION IS MAXIMUM INJ E2620 l

I NUREG-1101, Vol. 3 C-31 L

1ECTION RATE) E2630 560 FORMAT (1H ,15X,35H MAX. INJECTION OCCURS IN CELL IX = ,13,6H IY = E2635 1 ,I3) E2636 570 FORMAT (1H0,5X,47H*** WARNING *** DECREASE CRITERIA IN E 230-260) E2637 580 FORMAT (1H0,10X,63H* TIME INCREMENT FOR SOLUTE TRANSPORT EQUALS TIM E2638 1E STEP FOR FLOW *) E2639 END E2640-SUBROUTINE MOVE F 10 C* IBM REAL*8 TMRX,VPRM,HI,HR,HC HK,WT, REC,RECH, TIM,A0PT, TITLE F 20 C* IBM REAL*B XDEL,YDEL.S. AREA,SUMT,RH0,PARAM, TEST,TOL, PINT HMIN,PYR F 30 C* IBM REAL *8DXINV,DYINV,ARINV,PORINV F 32 INTEGER PTID F 35 LEVEL 2,N00EID,NPCELL,NP0LD, LIMB 0,1X0BS,1YOBS,THCK, PERM,TMWL,TM0BS N 825 LEVEL 2,TMRX,VPRM,HI,HR,HC,HK WT, REC,RECH, TIM,A0PT, TITLE,PART, CONC N 826 LEVEL 2,TMCN,VX,VY,CONINT,CNRECH,5UMC,VXBDY,VYBDY,PTID N 827 COMMON /PRMI/ NTIM,NPMP,NPNT,NITP,N,NX,NY,NP,NREC, INT,NNX,NNY,NUM0 F 40 IBS,NMOV IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N F 50 2PNCHV,NPDELC,ICHK F 61 COMMON /PRMC/ N0DEID(42,42).NPCELL(42,42),NP0LD(42,42), LIMB 0(500), F 65 IIX0BS(10),IY0BS(10) F 66 2,NDECAY,NSORB N 830 COMMON /HEDA/THCK(42,42), PERM (42,42),TMWL(10,50),TM0BS(50),ANFCTR F 90 COMMON /HEDB/ THRX(42,42,2),VPRM(42,42),HI(42,42),HR(42,42).HC(42, F 100 142),HK(42,42),WT(42,42), REC (42,42),RECH(42,42) TIM (100),A0PT(42),T F 110 21TLE(10),XDEL,YDEL,S AREA,SUMT, RHO,PARAM, TEST,TOL, PINT,HMIN,PYR F 120 COMMON /XINV/ DXINV,DYINV,ARINV,PORINV F 130 COMMON /CPMA/ PART(3,9000), CONC (42,42),TMCN(10,50),VX(42,42),VY(42 F 140 1,42),CONINT(42,42) CNRECH(42,42),POROS,SUMTCH, BETA,TIMV, STORM,STOR F 150 2MI.CMSIN,CMSOUT,FLMIN,FLMOT SUMIO,CELDIS,DLTRAT,CSTORM F 160 3,DCYLAM,BLKDEN,SRBRAT,SRBSAT,E9EALF,VOLDCY,VOLSRB,SRBDCY N 840 COMMON /CHMC/SUMC(42,42),VXBDY(42,42),VYBDY(42,42) F 170 COMMON /CHMP/ PTID(9000) F 175 C *************************************************************** F 190 WRITE (6,680) NMOV F 200 SUHTCH=SUMT-TIM (N) F 210 Fl=0.30 F 212 F2=1.0/3.0 F 214 IF (NPTPND.EQ.4) Fl=0.25 F 216 IF (NPTPND.EQ.9) Fl=F2 F 218 TF (NPTPND.EQ.8) F2=0.25 F 222 CONST1=TIMV*DXINV F 250 CONST2=TIMV*DYINV F 260 C ---MOVE PA.RTICLES "NMOV" TIMES--- F 270 NUREG-1101, Vol. 3 C-32

A DO 650 IM0V=1,NMOV F 280 10 NPTM=NP F 290 C ---MOVE EACH PARTICLE--- F 300 00 590 IN=1,NP F 310 IF (PART(1,IN).EQ.0.0) GO TO 590 F 320 KFLAG=0 F 330 C *************************************************************** F 340 C ---COMPUTE OLD LOCATION--- F 350 20 X0LD=PART(1,IN)

F 400 IX=X0LD+0.5 F 410 IFLAG=1 F 420 IF (PART(2,IN).GE.0.0) GO TO 30 F 430 IFLAG=-1 F 440 PART(2,IN)=-PART(2,1N) F 450 30Y0LD=PART(2,IN) F 460 IY=YOLD+0.5 F 470 IF (THCK(IX,IY).EQ.0.0) GO TO 590 F 482 C *************************************************************** F 490 C

---COMPUTE NEW LOCATION AND LOCATE CLOSEST N0DE--- F 500 C ---LOCATE NORTHWEST CORNER--- F 510 IVX=X0LD F 520 IVY =YOLD F 530 IXE=IVX+1 F 540 IYS= IVY +1 F 550 C *************************************************************** F 560 C ---LOCATE QUADRANT, VEL. AT 4 CORNERS, CHECK FOR BOUNDARIES--- F 570 CELDX=X0LD-IX F 580 CELDY=Y0LD-IY F 590 ICD =9 F 595 IF (CELDX.EQ.0.0.AND.CELDY.EQ.0.0) GO TO 280 F 600 IF (CELDX.GE.0.0.0R.CELDY.GE.0.0) GO TO 70 F 610 C . ---PT. IN NW QUADRANT--- F 620 VXNW=VXBDY(IVX, IVY)

F 630 VXNE=VX(IXE IVY) F 640 VXSW=VXBDY(IVX,IYS)

F 650 VXSE=VX(IXE,IYS)

F 660 VYNW=VYBDY(IVX, IVY)

F 670 VYNE=VYBDY(IXE, IVY)

F 680 VYSW=VY(IVX,IYS) F 690 VYSE=VY(IXE,IYS)

F 700 100=1 F 705 IF(THCK(IVX, IVY).EQ.0.0)GOTO50 F 710 IF(REC (IXE, IVY).EQ.0.0.AND.VPRM(IXE, IVY).LT.O.09)GOTO40 F 720 NUREG-1101, Vol. 3 C-33

VXNE=VXNW F 730 40 IF (REC (IVX,1YS).EQ.0.0.AND.VPRM(IVX,1YS).LT.0.09) GO TO 50 F 740 VYSW=VYNW F 750 50 IF (REC (IXE IYS).EQ.0.0.AND.VPRM(IXE,1YS).LT.0.09) GO TO 270 F 760 IF (THCK(IVX,IYS).EQ.0.0) GO TO 60 F 770 IF (THCK(IXE+1,IYS).GT.0.0) VXSE=VXSW F 782 60IF(THCK(IXE, IVY).EQ.0.0)GOTO270 F 790 IF(THCK(IXE,IYS+1).GT.0.0)VYSE=VYNE F 802 GO TO 270 F 810 C

F 820 F 830 70 IF (CELDX.LE.0.0.0R.CELDY.GE.0.0) GO TO 130 F 840 C ---

PT. IN NE QUADRANT---

80VXNW=VX(IVX, IVY) F 850 VXNE=VXBDY(IVX, IVY) F 860 VXSW=VX(IVX,IYS) F 870 VXSE=VXBDY(IVX,1YS F 880 VYNW=VYBDY(IVX, IVY F 890 VYNE=VYBDY(IXE, IVY F 900 VYSW=VY(IVX,IYS) F 910 F 920 VYSE=VY(IXE IYS)

F 925 ICD =2 F 930 IF (CELDX.EQ.0.0) GO TO 120 F 940 IF (THCK(IXE, IVY).EQ.0.0) GO TO 100 IF (REC (IVX, IVY).EQ.0.0.AND.VPRM(IVX, IVY).LT.0.09) GO TO 90 F 950 VXNW=VXNE F 960 90 IF (REC (IXE,1YS).EQ.0.0.AND.VPRM(IXE,IYS).LT.0.09) GO TO 100 F 970 F 980 VYSE=VYNE 100 IF (REC (IVX,IYS).EQ.0.0.AND.VPRM(IVX,1YS).LT.0.09) GO TO 270 F 990 IF (THCK(IXE,IYS).EQ.0.0) GO TO 110 F1000 IF(THCK(IVX-1,1YS).GT.0.0)VXSW=VXSE F1012 110 IF (THCK(IVX, IVY).EQ.0.0) GO TO 270 F1020 IF(THCK(IVX,IYS+1).GT.0.0)VYSW=VYNW F1032 F1040 GO TO 270 120 IF (REC (IVX,IYS).EQ.0.0.AND.VPRM(IVX,1YS).LE.0.09) GO TO 270 F1050 IF (THCK(IVX, IVY).EQ.0.0) GO TO 270 F1060 IF (THCK(IVX,1YS+1).GT.0.0) VYSW=VYNW F1072 F1080 GO TO 270 F1090 C

F1100 130 IF (CELDY.LE.0.0.0R.CELDX.GE.0.0) GO TO 190 F1110 C ---

PT. IN SW QUADRANT---

140'VXNW=VXBDY(IVX, IVY) F1120 VXNE=VX(IXE, IVY) F1130 VXSW=VXBDY(IVX,1YS) F1140 NUREG-1101, Vol. 3 C-34

VXSE=VX(IXE,IYS F1150 VYNW=VY(IVX, IVY F1160 VYNE=VY(IXE, IVY F1170 VYSW=VYBDY(IVX, IVY) F1180 VYSE=VYBDY(IXE, IVY) F1190 ICD =3 F1195 IF CELDY.EQ.0.0)GOTO180 F1200 IF THCK(IVX,1YS).EQ.0.0) GO TO 160 F1210 IF REC (IVX, IVY).EQ.0.0.AND.VPRM(IVX, IVY).LT.0.09) GO TO 150 F1220 VYNW=VYSW F1230 150 IF (REC (IXE,IYS).EQ.0.0.AND.VPRM(IXE,IYS).LT.0.09) GO TO 160 F1240 VXSE=VXSW F1250 160 IF (REC (IXE, IVY).EQ.0.0.AND.VPRM(IXE, IVY).LT.0.09) GO TO 270 F1260 IF (THCK IVX, IVY).EQ.0.0) GO TO 170 F1270 IF (THCK IXE+1, IVY).GT.O.0) VXNE=VXNW F1282 170 IF (THCK IXE,IYS).EQ.0.0) GO TO 270 .F1290 l IF(THCKIXE, IVY-1).GT.O.0)VYNE=VYSE F1302 GO TO 270 F1310 180 IF (REC (IXE, IVY).EQ.0.0.AND.VPRM(IXE, IVY).LE.0.09) GO TO 270 F1320 IF (THCK(IVX, IVY).EQ.0.0) GO TO 270 F1330 IF (THCK(IXE+1, IVY).GT.0.0) VXNE=VXNW .F1342 GO TO 270 F1350 C

F1360 190 IF (CELDY.LE.0.0.0R.CELDX.LE.0.0) GO TO 260 F1370

, C . ---PT. IN SE QUADRANT--- F1380 200 VXNW=VX(IVX, IVY) F1390 VXNE=VXBDY(IVX, IVY)

' F1400 VXSW=VX(IVX,IYS)

F1410 3 VXSE=VXBDY(IVX,1YS)

F1420 l VYNW=VY(IVX, IVY)

F1430 VYNE=VY(IXE, IVY)

• F1440 VYSW=VYBDY(IVX, IVY) F1450 i VYSE=VYBDY(IXE, IVY)

F1460 ICD =4 F1465 IF (CELDY.EQ.0.0) GO TO 240 F1470 IF(CELDX.EQ.0.0)GOTO250 F1480 IF (THCK(IXE,IYS).EQ.0.0) GO TO 220 F1490 IF (REC (IXE, IVY).EQ.0.0.AND.VPRM(IXE, IVY).LT.0.09) GO TO 210 F1500

, VYNE=VYSE F1510

.: 210 IF (REC (IVX,1YS).EQ.0.0.AND.VPRM(IVX,1YS).LT.0.09) GO TO 220 F1520 VXSW=VXSE F1530 220 IF (REC (IVX, IVY).EQ.0.0.AND.VPRM(IVX, IVY) LT.0.09) GO TO 270 F1540 IF (THCK(IXE. IVY).EQ.0.0) GO TO 230 F1550

-NUREG-1101, Vol. 3 C-35

IF- THCK(IVX-1, IVY).GT.0.0) VXNW=VXNE F1562 230 IF THCK(IVX,1YS).EQ.0.0) GO TO 270 F1570

'IF THCK(IVX, IVY-1).GT.O.0)VYNW=VYSW F1582 GO TO 270 F1590 240 IF REC (IVX, IVY).EQ.0.0.AND.VPRM(IVX, IVY).LE.0.09) GO TO 270 -F1600 IF THCK(IXEIVY).EQ.0.0)GOTO270 F1610 IF. THCK(IVX-1, IVY).GT.0.0) VXNW=VXNE F1622 GO TO 270 F1630 250 IF (REC (IVX, IVY).EQ 0.0.AND.VPRM(IVX, IVY).LE.0.09) GO TO 270 F1640 IF (THCK(IVX,1YS).EQ.0.0) GO TO 270 F1650-IF(THCK(IVX, IVY-1).GT.0.0)VYNW=VYSW F1662 F1670

~

GO TO 270 C F1680 260 IF (CELDX.EQ.0.0.AND.CELDY.LT.0.0) GO TO 80 F1690 IF (CELDX.LT.0.0.AND.CELDY.EQ.0.0) GO TO 140 F1700 IF(CELDX.GT.0.0.AND.CELDY.EQ.0.0)GOT0,200 F1710 IF (CELDX.EQ.0.0.AND.CELDY.GT.0.0) GO TO 200 F1720 WRITE (6,690)IN,IX,1Y F1730 270 CONTINUE F1740 C --- CHECK FOR ADJACENT NO-FLOW BOUNDARIES--- F1741A GO T0 (1270,1275,1280,1285,1290) ICD F1741B G010 1290 F1741C 1270 IF (THCK(IXE. IVY .EQ.0.0) GO TO 1272 F1742A IF (THCK(IVX,IYS .EQ.0.0) GO TO 1273 F1742B IF (THCK(IVX, IVY .EQ.0.0) GO TO 1274 F1742C GO TO 1290 F1742D 1272 VXNE=VXSE F1742E IF (THCK(IVX,1YS).GT.0.0) GO TO 1274 F1742F 1273 VYSW=VYSE F1742G 1274 VXNW=VXSW F1742H

! VYNW=VYNE F17421 l GO TO 1290 F1742J 1275 IF THCK IVX, IVY).EQ.0.0) GO TO 1277 F1744A IF THCK IXE,1YS).EQ.0.0) GO TO 1278 F1744B IF THCK IXE, IVY).EQ.0.0) GO TO 1279 F1744C GO TO 1290 F1744D 1277 VXNW=VXSW F1744E IF (THCK(!XE,IYS).GT.0.0) GO TO 1279 F1744F 1278 VYSE=VYSW F1744G 1279 VXNE=VXSE F1744H VYNE=VYNW F17441 GO TO 1290 F1744J 1280 IF (THCK(IXE,1YS).EQ.0.0) GO TO 1282 F1746A NUREG-1101, Vol. 3 C-36

IF(THCK(IVX, IVY).EQ.0.0)GOTO1283 F17468 IF (THCK(IVX,IYS).EQ.0.0) GO TO 1284 F1746C GO TO 1290 F1746D 1282 VXSE=VXNE F1746E IF (THCK(IVX, IVY).GT.0.0) GO TO 1284 F1746F 1283 VYNW=VYNE F1746G 1284 VXSW=VXNW F1746H VYSW=VYSE F1746I GO TO 1290 F1746J 1285 IF (THCK(IVX,IYS).EQ.0.0)-GO TO 1287 F1748A IF (THCK(IXE IVY).EQ.0.0) GO TO 1288 F1748B IF (THCK(IXE,1YS).EQ.0.0) GO TO 1289 F1748C GO TO 1290. F1748D 1287 VXSW=VXNW F1748E IF (THCK(IXE, IVY).GT.0.0) GO TO 1289 F1748F 1288 VYNE=VYNW F1748G 1289.VYSE=VYSW F1748H VXSE=VXNE F17481 1290 CONTINUE F1749A C '**************************************************************** F1750 C ---BILINEAR INTERPOLATION--- F1760 CELXD=X0LD-IVX F1770 CELDXH=AMOD(CELXD,0.5) F1780 CELDX=CELDXH*2.0 F1790 CELDY=Y0LD-IVY F1800 C **************************************************************** F1810 C ---X VELOCITY--- F1820 VXN=VXNW*(1.0-CELDX)+VXNE*CELDX F1830 VXS=VXSW*(1.0-CELDX)+VXSE*CELDX F1850 XVEL=VXN*(1.0-CELDY)+VXS#CELDY F1870

C ---Y VELOCITY--- F1900 i CELDYH=AMOD(CELDY,0.5) l F1910

.CELYD=CELDYH*2.0 F1921

~VYW=VYNW*(1.0-CELYD)+VYSW*CELYD F1931 VYE=VYNE*(1.0-CELYD)+VYSE*CELYD F1951 YVEL=VYW*(1.0-CELXD)+VYE*CELXD F1970 C

F2000 GO TO 290 r2010 280XVEL=VX(IX,IY) F2020 YVEL=VY(IX,IY)

F2030 290 DISTX=XVEL*CONST1 F2040 DISTY=YVEL*CONST2 F2050 C **************************************************************** F2060 C ---BOUNDARY CONDITIONS--- F2070 NUREG-1101, Vol. 3 C-37

TEMPX=X0LD+DISTX F2080 TEMPY=YOLD+DISTY F2090 INX=TEMPX+0.5 F2100 INY=TEMPY+0.5 F2110 IF (THCK(INX,INY).GT.0.0) GO TO 330 F2120 C **************************************************************** F2130 C ---X B0UNDARY--- F2140 IF (THCK(INX,IY).EQ.0.0) GO TO 300 F2150 PART(1,IN)=TEMPX F2160 GO TO 310 F2170 300 BEYON=TEMPX-IX F2180 IF(BEYON.LT.0.0)BEYON=BEYON+0.5 F2190 IF(BEYON.GT.0.0)BEYON=BEYON-0.5 F2200 PART(1,IN)=TEMPX-2.0*BEYON F2210 INX=PART(1,IN)+0.5 F2220 TEMPX=PART(1,1N) F2230 C **************************************************************** F2240 C ---Y B0UNDARY--- F2250 310 IF (THCK(INX,1NY).EQ.0.0) GO TO 320 F2260 PART(2,1N)=TEMPY F2270 GO TO 340 F2280 C **************************************************************** F2290 320 BEYON=TEMPY-IY F2300 IF (BEYON.LT.0.0) BEYON=BEYON+0.5 F2310 IF (BEYON.GT.O.0) BEYON=BEYON-0.5 F2320 PART(2,1N)=TEMPY-2.0*BEYON F2330 INY=PAF,T(2,IN)+0.5 F2340 TEMPY=PART(2,1N) F2350 GO TO 340 F2360 330PART(1,1N)=TEMPX F2370 PART(2,IN)=TEMPY F2380 340 CONTINUE F2390 C *************************************************************** F2400 C ---SUM CONCENTRATIONS AND COUNT PARTICLES--- F2410 SUMC(INX,1NY)=SUMC(INX,1NY)+PART(3,1N) F2420 NPCELL(INX,INY)=NPCELL(INX,INY)+1 F2430 C **************************************************************** F2440 C ---CHECK FOR CHANGE IN CELL LOCATION--- F2450 IF (IX.EQ.INX.AND.IY.EQ.INY) GO TO 580 F2460 C ---CHECK FOR CONST.-HEAD CDY. OR SOURCE AT OLD LOCATION--- F2470 IF(REC (IX,1Y).LT.0.0)GOTO350 F2480 IF (REC (IX,1Y).GT.0.0) GO TO 360 F2490 IF (VPRM(IX,IY).LT.0.09) GO TO 540 F2500 NUREG-1101, Vol. 3 C-38

i IF(WT(IX,IY).GT.HK(IX,IY))GOTO350 F2510 IF (WT(IX,IY).LT.HK(IX,IY)) GO TO 360 F2520 GO TO 540 F2530 C **************************************************************** F2540 C ---CREATE NEW PARTICLES AT B0UNDARIES--- F2550 350 IF (IFLAG.GT.0) GO TO 550 F2560 KFLAG=1 F2570 360 00 370 IL=1,500 F2580 IF (LIMB 0(IL).EQ.0) GO TO 370 F2590 IP= LIMB 0(IL) F2600 IF (IP.LT.IN) GO TO 380 F2610 370 CONTINUE F2620 C **************************************************************** F2630 C ---GENERATE NEW PARTICLE--- F2640 IF (NPTM.EQ.NPMAX) GO TO 600 F2650 NPTM=NPTM+1 F2660 IP=NPTM F2670 GO TO 390 F2680 380 LIMB 0(IL)=0 F2690 C

F2700 390 IF (KFLAG.EQ.0) GO TO 398 F2705 ITEM =PTID(IN) F2845 GO TO 399 F2855 398 SUMC(IX,IY)=SUMC(IX,IY)+ CONC (IX,IY) F2865 NPCELL(IX,IY)=NPCELL(IX,IY)+1 F2875 j IF NP0LD(IX,1Y).GT.0) NP0LD(IX,1Y)=NP0LD(IX,1Y)-1 F2885 t

IF IFLAG.GT.0) GO TO 441 F2895 IF KFLAG.EQ.0) GO TO 441 F2899 l 399 GO T0 (401,411,421,431,441,451,461,471,481), ITEM F2905 l GO TO 441 F2915 l 401 PART(1,IP)=IX-F1 F2925 PART(2,IP)=IY-F1 F2935 PTID(IP)=1 F2935 GO TO 530 F2955 411 PART(1,IP)=IX-F1 F2965 PART(2,IP)=IY+F1 F2975 PTID(IP)=2 F2985 GO TO 530 421PART(1,IP)=IX+F1 F2995 PART(2,IP)=IY-F1 F3005 F3015 PTID(IP)=3 F3025 GO TO 530 431 PART(1,IP)=IX+F1 F3035 F3045 NUREG-1101, Vol. 3 C-39

l i

i l

PART(2,IP)=IY+F1 F3055 PTID(IP)=4 F3065 GO TO 530 F3075 441PART1,IP)=IX F3085 PART 2,IP)=IY F3095 PTID IP)=5 F3105 GO TO 530 F3115 451 PART 1,IP)=IX-F2 F3125 PART 2,IP)=IY F3135 PTID IP)=6 F3145 GO TO 530 F3155 461PART(1,IP)=IX F3165 PART(2,IP)=IY-F2 F3175 PTID(IP)=7 F3185 GO TO 530 F3195 471 PART 1,IP)=IX+F2 F3205 PART 2,IP)=IY F3215 PTID IP)=8 F3225 GO TO 530 F3235 481PART(1,IP)=IX F3245 PART(2,IP)=IY+F2 F3255 PTID(IP)=9 F3265 C F3510 530PART(2,IP)=-PART(2,IP) F3520 PART(3,IP)= CONC (IX,IY) F3530 C **************************************************************** F3550 C ---CHECK FOR DISCHARGE B0UNDARY AT NEW LOCATION--- F3560 540 IFLAG=1.0 F3570 550 IF (VPRM(INX,IIIY).GT.0.09.AND.WT(INX,1NY).LT.HK(INX,INY)) GO TO 56 F3580 10 F3590 IF (REC (INX,INY).GT.0.0) GO TO 560 F3600 GO TO 590 F3610 C **************************************************************** F3620 C ---PUT PT. IN LIMB 0 IF PT. DENSITY NOT INCREASED--- F3625 560 IF (NPOLD(INX,1NY).LE.0) GO TO 590 F3635 l PART(1,1N)=0.0 F3645 PART(2,1N)=0.0 F3650 PART(3,1N)=0.0 F3660 SUMC(INX,INY)=SUMC(INX,INY)-CONC (INX,INY) F3662 NPCELL(INX,1NY)=NPCELL(INX,INY)-1 F3664 NPOLD(INX,INY)=NPOLD(INX,INY)-1 F3666 DO 570 ID=1,500 F3670 F3680 IF (LIMB 0(ID).GT.0) GO TO 570 NUREG-1101, Vol. 3 C-40

LIMB 0(ID)=lN F3690 GO TO 590 F3700 570 CONTINUE F3710 C F3720 580 IF (IFLAG.LT.0) PART(2,IN)=-TEMPY F3730 590 CONTINUE F3750 C ---END OF LOOP--- F3760 C **************************************************************** F3770 GO TO 620 F3780 C ---RESTART MOVE IF PT. LIMIT EXCEEDED--- F3790 600 WRITE (6,700) IMOV,IN F3800 TEST =100.0 F3810 CALL GENPT F3820 D0 610 IX=1,NX F3830 00 610 IY=1,NY F3840 SUMC(IX,IY)=0.0 F3850 610 NPCELL(IX,IY)=0 F3860 TEST =0.0 F3870 GO TO 10 F3880 C **************************************************************** F3890 620 SUMTCH=SUMTCH+TIMV F3900 C ---ADJUST NUMBER OF PARTICLES--- F3910 NP=NPTM F3920 WRITE (6.670) NP,IMOV F3930 C *****************************************************"********** F3940 CALL CNCON F3950 C **************************************************************** F3960 C ---STORE OBS. WELL DATA FOR STEADY FLOW PROBLEMS--- F3970 IF (S.GT.0.0) GO TO 640 F3980 IF (NUMOBS.LE.0) GO TO 640 F3990 J=M00(IMOV.50) F4000 IF (J.EQ.0) J=50 F4010 TMOBS(J)=SUMTCH F4020 00 630 I=1,NUMOBS F4030 TMWL(I,J)=HK(IX0BS(I),IY0BS(I)) F4C40 630 TMCN(I,J)= CONC (IX0BS(I),IY0BS(I)) F4050 C ---PRINT CHEMICAL OUTPUT--- F4060 IF (M00(IM0V,50).EQ.0) IPRNT = 1 F4065 640 IF (IMOV.GE.NMOV) GO TO 660 F4070 IF (MOD (IMOV,NPNTMV).EQ.0) IPRNT = -1 F4082 650 IF (IPRNT.NE.0) CALL CHM 0T F4085 C **************************************************************** F4090 660 RETURN F4100 C **************************************************************** F4110 NUREG-1101, Vol. 3 C-41

i C F4120 l C F4130 C F4140 670 FORMAT (1HO,2X,2HNP,7X,2H= .8X,14,10X,11HIMOV = ,8X,I4) F4150 680 FORMAT (1H0,10X,61HN0. OF PARTICLE MOVES REQUIRED TO COMPLETE THIS F4160 1 TIME STEP = ,14//) F4170 690 FORMAT (1H0,5X,53H*** WARNING *** QUADRANT NOT LOCATED FOR PT. F4180 1 N0. ,15,11H , IN CELL ,214) F4190 700 FORMAT (1H0,5X,17H *** NOTE *** 10X,23HNPTM.EQ.NPMAX

, --- IMOV= F4200 1,14,5X,8HPT. N0.=,14,5X,10HCALL GENPT/) F4210 END F4220-SUBROUTINE CNCON G 10 C* I BM REAL*8 TMRX ,V PRM ,H I ,HR ,HC , H K ,WT , REC , RECH ,T I M , A0PT ,T I TLE G 20 C* IBM REAL*8 XDEL,YDEL,S. AREA,SUMT, RHO,PARAM, TEST,TOL, PINT,HMIN,PYR G 30 C* IBM REAL *8DXINV,DYINV,ARINV,PORINV G 35 C* IBM REAL*8 FLW G 40 LEVEL 2,N0DEID,NPCELL,NPOLD, LIMB 0,IX0BS.IY0BS,THCK, PERM,TMWL,TMOBS N 842 LEVEL 2,TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT, TITLE.PART, CONC N 843 LEVEL 2,TMCN,VX,VY,CONINT,CNRECH, DISP,50MC VXBDY,VYBDY N 844 COMMON /PRMI/ NTIM,NPMP,NPNT,NITP N,NX,NY,NP NREC, INT,NNX,NNY,NUM0 G 50 IBS,NM0V,IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N G 60 2PNCHV,NPDELC,ICHK G 71 COMMON /PRMC/ N0DEID(42,42),NPCELL(42,42),NPOLD(42,42), LIMB 0(500), G 75 IIX0BS(10),IY0BS(10) G 76 2,NDECAY,NSORB N 850 COMMON /HEDA/THCK(42,42), PERM (42,42),TMWL(10,50),TM0BS(50),ANFCTR G 100 COMMON /HEDB/ TMRX(42,42,2),VPRM(42,42),HI(42,42),HR(42,42),HC(42, G 110 142),HK(42,42),WT(42,42), REC (42,42),RECH(42,42), TIM (100) A0PT(42),T G 120 21TLE(10),XDEL,YDEL,S, AREA,SUMT,RH0,PARAM TEST,TOL, PINT,HMIN,PYR G 130 COMMON /XINV/ DXINV,DYINV,ARINV,PORINV G 140 COMMON /CHMA/ PART(3,9000), CONC (42,42),TMCN(10,50),VX(42,42),VY(42 G 150 ,

1,42),CONINT(42,42),CNRECH(42,42),POROS,SUMTCH, BETA,TIMV, STORM,STOR G 160 2MI CMSIN. CMS 0VT,FLMIN,FLMOT, SUM 10,CELDIS,DLTRAT,CSTORM G 170 3,0CYLAM,BLKDEN,SRBRAT SRBSAT,SRBALF,VOLDCY,VOLSRB,SRBDCY N 860 COMMON /DIFUS/ DISP (42,42,4) G 180 COMMON /CHMC/ SUMC(42,42),VXBDY(42,42),VYBDY(42,42) G 190 DIMENSION CNCNC(42,42), CN0LD(42,42) G 200 C *************************************************************** G 210 ITEST=0 G 220 NZER0=0 N 870 STORM =0.0 N 880 00 10 IX=1,NX G 230 00 10 IY=1,NY G 240 NUREG-1101, Vol. 3 C-42

CN0LD(IX,IY)= CONC (IX,IV) G 250 IF (THCK(IX,1Y).EQ.0.0) GO TO 10 N 890 IF (NPCELL(IX,1Y).GT.O.0) N 900 1 CONC (IX,1Y)= 0.5*(CONC (IX,IY) + SUMC(IX,IY)/(1.0*NPCELL(IX,IY))) N 910 10 CNCNC(IX,IY)=0.0 G 260 l

.TVA= AREA *TIMV G 290 '

ARPOR= AREA *POROS G 300 C *************************************************************** G 310 C ---CONC. CHANGE FOR TIMV DUE TO: G 320 ,

C RECHARGE, PUMPING, LEAKAGE, DECAY , SORBTION--- G 330 '

20 DO 60 IX=1,NX G 350 l DO 60 IY=1,NY G 360 IF (THCK(IX,IY).EQ.0.0) GO TO 60 G 370 CSTAR = CONC (IX,IY) N 920 l IF (NPCELL(IX,IY).GT.0) CSTAR = SUMC(IX,IY)/(1.0*NPCELL(IX,IY)) N 930 CKM1 = 2.0* CONC (IX,IY) - CSTAR N 940 DIST = 0.0 N 950 IF (NSORB.LT.1) GO TO 50 N 960 IF (NSORB.GT.1) GO TO 30 N 970 C*** ****************** LINEAR SORBTION*************** N 980 DIST = SRBRAT*BLKDEN N 990 GO TO 50 N1000 30 IF (NSORB.GT.2) GO TO 40 N1010 C*** ******************LANGMUIR SORBTION************* N1020 DIST = BLKDEN*SRBRAT*SRBSAT/(1.0+SRBRAT*CKM1)**2.0 N1030 GO TO 50 N1040 40 IF (SRBALF.EQ.0.0) GO TO 50 N1050 C*** ******************FREUNDLICH S0RBTION*********** N1060 LOGCON = -23.0 N1070 IF (CKM1.GT.1.0E-10) LOGCON = ALOG(CKM1) N1080 SRBEXP = (SRBALF-1.0)*LOGCON N1090 IF (SRBEXP.GT.23.0) SRBEXP = 23.0 N1100 DIST = BLKDEN*SRBALF*SRBRAT*EXP(SRBEXP) N1110 50 EQFCT1=TIMV/((1.0+DIST)*THCK(IX,IY)) N1120 EQFCT2=EQFCT1/POR0S N1130 DELC = 0.0 N1140 DELCR = CNRECH(IX,IY) - CONC (IX,IY) N1150 C ---ACCOUNT FOR BOUNDARY FLOW--- N1160 FLW = VPRM(IX,IY)*(WT(IX,IY)-HK(IX,IY)) N1170 C ---MASS IN B0UNDARY DURING TIME STEP--- N1180 IF (FLW.GT.0.0) FLMIN=FLMIN+FLW*CNRECH(IX,IY)*TVA N1190 C ---MASS OUT DURING TIME STEP--- N1200 IF (FLW.LT.0.0) FLMOT=FLMOT+FLW* CONC (IX,IY)*TVA N1210 s

NUREG-1101, Vol. 3 C-43

IF (FLW.GT.0.0) DELC = DELC + FLW*DELCR .

N1220 C ---ACCOUNT FOR MASS PUMPED IN, OUT, RECHARGED, + DISCHARGED--- N1230 RATE = REC (IX,1Y)*ARINV N1240 IF(RATE.GT.0.0 CMSOUT=CMSOUT+ RATE

• CONC (IX,IY)*TVA N1250 IF(RATE.LT.O.0 CMSIN=CMSIN+ RATE *CNRECH(IX,IY)*TVA N1260 IF (RATE.LT.O.0 DELC.='DELC - RATE *DELCR N1270 t QRECH = RECH(IX,1Y) N1280 IF QRECH.GT.O.0) CMS 00T= CMS 00T+QRECH* CONC (IX,1Y)*TVA N1290

- IF QRECH.LT.O.0) CMSIN=CMSIN+QRECH*CNRECH(IX,IY)*TVA .N1300 IF QRECH.LT.0.0) DELC = DELC - QRECH*DELCR N1310 DELDCY = 0.0 N1320 J- IF(NDECAY.GT.0 DELDCY = N1330 1 THCK(IX,1Y*POROS* CONC (IX,IY)*(EXP(-DCYLAM*TIMV)-1.0)/TIMV N1340 IF NDECAY.GT.0 VOLDCY = VOLDCY + DELDCY*TVA N1350

~

IF NDECAY.GT.O.AND.NSORB.GT.0) SRBDCY = SRBDCY + DIST*DELDCY*TVA N1360

'IF NDECAY.GT.0.AND.NSORB.GT.0) DELDCY = DELDCY + DIST*DELDCY N1370 '

DELC = DELC + DELDCY N1380

1. DELC = DELC*EQFCT2 N1390

' N1400 CNCNC(IX,IV) = CNCNC(IX,IY) + DELC i C ---CONC. CHANGE DUE TO DISPERSION--- N1410

C ---DISPERSION WITH TENSOR COEFFICIENTS--- N1420 DELCD = 0.0 N1430 J

i IF(BETA.EQ.0.0)GO.T055 N1440

G 650 X1=DISPIX,IY,1)*(CONC (IX+1,IY)-CONC X2=DISPIX-1,IY,1)*(CONC (IX-1,1Y)-CONC (IX,IY) (IX,1Y))) G 660 l Yl=DISPIX,1Y,2)*(CONC (IX,IY+1)-CONC (IX,1Y)) G 670 4

Y2=DISPIX,IY-1,2)*(CONC (IX,IY-1)-CONC (IX,1Y)) G 680 1 XX1= DISP (IX,1Y,3)*(CONC (IX,IY+1)+ CONC (IX+1,1Y+1)-CONC (IX,1Y-1)-CON G 690 1C(IX+1,IY-1)) G 700 XX2= DISP (IX-1,IY,3)*(CONC (IX,IY+1)+ CONC (IX-1,1Y+1)-CONC (IX,1Y-1)-C G 710 10NC(IX-1,IY-1)) G 720

YYl= DISP (IX,IY,4)*(CONC (IX+1,IY)+ CONC (IX+1,IY+1)-CONC (IX-1,IY)-CON G 730 1C(IX-1,IY+1)) G 740 YY2= DISP (IX,1Y-1,4)*(CONC (IX+1,IY)+ CONC (IX+1,1Y-1)-CONC (IX-1,IY)-C G 750

, 10NC(IX-1,IY-1)) G 760

! DELCD = EQFCT1*(X1+X2+Yl+Y2+XX1-XX2+YY1-YY2)*(1.0+DIST) N1450

55 CNCNC(IX,IY) = CNCNC(IX,1Y) + DELCD N1460 i C ---COMPUTE MASS OF SOLUTE IN STORAGE--- N1470 i STORM = STORM + (CSTAR+DELC+DELCD)*THCK(IX,1Y)*ARPOR N1480 C ---COMPUTE CHANGE OF SORBED SOLUTE--- N1490 N1500 l IF(NSORB.GT.0)VOLSRB=VOLSRB+

1 DIST*THCK(IX,1Y)*ARPOR*(DELC+DELCD+CSTAR-CKM1) N1510 1

j 60 CONTINUE N1520 1

.1 l

l NUREG-1101, Vol. 3 C-44 l

00 90 IX=1,NX G 850 DO 90 IY=1,NY G 860 IF (THCK(IX,IY).EQ.0.0) GO TO 90 G 870 APC=NPCELL(IX,IY) G 880 IF(APC.GT.0.0)GOTO80 G 890 IF(REC (IX,IY).NE.0.0.0R.VPRM(IX,IY).GT.0.09)GOTO90 G 900 NZER0=NZER0+1 G 910 GO TO 90 G 920 80 CONC (IX,1Y)=SUMC(IX,IY)/APC G 930 90 CONTINUE G 940 C ---CHECK NUMBER OF CELLS VOID OF PTS.--- G 950 IF (NZERO.GT.0) WRITE (6,290) NZERO,IMOV G 960 IF (NZERO.LE.NZCRIT) GO TO 110 N1530 TEST =99.0 G 980 WRITE (6,300) G 990 WRITE (6.320) G1000 D0 100 IY=1,NY G1010 100 WRITE (6,330)(NPCELL(IX,IY),1X=1,NX) G1020 GO TO 110 N1540 C **************************************************************** G1040 C ---CHANGE CONCENTRATIONS AT N0 DES--- G1050 110 D0 130 IX=1,NX G1060 00 130 IYal,NY G1070 IF (THCK(IX,IY).EQ.0.0) GO TO 120 G1080 CONC (IX,IY)= CONC (IX,IY)+CNCNC(IX,IY) G1090 SUMC(IX,IY)=0.0 G1110 IF (CONC (IX,IY).LE.0.0) GO TO 130 G1120 CNCPCT=CNCNC(IX,IY)/ CONC (IX,1Y) G1130 SUNC(IX,IY)=CNCPCT G1140 GO TO 130 G1150 l 120 IF (CONC (IX,1Y).GT.0.0) WRITE (6,310) IX,1Y, CONC (IX,1Y) G1160 i CONC (IX,IY)=0.0 G1170 l

130 CONTINUE G1180 C **************************************************************** G1190 C ---CHANGE CONCENTRATION OF PARTICLES--- G1200 D0 180 IN=1,NP G1210 IF (PART(1,IN).EQ.0.0) GO TO 180 G1220 INX= ABS (PART(1,IN))+0.5 G1230 INY= ABS (PART(2,IN))+0.5 G1240 C ---UPDATE CONC. OF PTS. IN SINK / SOURCE CELLS--- G1250 IF (REC (INX,INY).NE.0.0) GO TO 140 G1260 IF(VPRM(INX,INY).LE.0.09)GOTO150 G1270 140 PART(3,IN)= CONC (INX,INY) G1280 NUREG-1101, Vol. 3 C-45

GO TO 180 G1290 150 IF (CNCNC(INX,1NY).LT.0.0) GO TO 170 G1300 160PART(3,1N)=PART(3,IN)+CNCNC(INX,INY) G1310 GO TO 180 G1320 170IF(CONC (INX,INY).LE.0.0)GOTO160 G1330 IF (SUNC(INX,1NY).LT.-l.0) GO TO 160 G1340 PART(3,1N)=PART(3,1N)+PART(3,1N)*SUMC(INX,INY) G1350 180 CONTINUE G1360 WRITE (6,280) TIM (N),TIMV,SUMTCH G1370 C **************************************************************** G1380 C ---COMPUTE MASS BALANCE FOR SOLUTE--- G1390 D0 270 IX=1,NX G1420 00 270 IY=1,NY G1430 IF (THCK(IX,1Y).EQ.0.0) GO TO 270 G1440 SUMC(IX,1Y)=0.0 G1450 NPOLD(IX,IY)=NPCELL(IX,IY) N1550 NPCELL(IX,1Y)=0 G1684 270 CONTINUE G1690 C **************************************************************** G1700 C ---COMPUTE CHANGE IN MASS OF SOLUTE STORED--- G1710 CSTORM= STORM-STORMI G1720 SUMIO=FLMIN+FLMOT-CMSIN-CM50VT G1730 C **************************************************************** G1740 C ---REGENERATE PARTICLES IF "NZCRIT" EXCEEDED--- G1750 IF(TEST.GT.98.0)CALLGENPT G1760 TEST =0.0 G1770 IF (NSORB.GT.1) CALL VELNEW (CN0LD) N1560 C **************************************************************** G1780 RETURN G1790 C **************************************************************** G1800 C G1810 C G1820 C G1830 280 FORMAT (3H ,11HTIM(N) = .1P1E12.5,10X,11HTIMV = ,1 PIE 12.5, G1840 110X,9HSUMTCH = ,1P1E12.5) G1850 290 FORMAT (1H0,5X,40HNUMBER OF CELLS WITH ZER0 PARTICLES = ,14,5X,9 G1860 1HIMOV =

14/) G1870 300 FORMAT (1H0,5X,44H*** NZCRIT EXCEEDED ---

CALL GENPT ***/) G1880 310 FORMAT (1H .5X,37H*** CONC.GT.0.AND.THCK.EQ.0 AT N0DE = ,2I4,4X,7HC G1890 10NC = ,1P1E10.4,4H ***) G1900 320 FORMAT (IH0,2X,6HNPCELL/) G1910 G1920 330 FORMAT (1H .4X.20I3)

END G1930-NUREG-1101, Vol. 3 C-46

SUBROUTINE OUTPT H 10 C* IBM REAL*8 TMRX,VPRM,HI,HR,HC.HK,WT, REC,RECH TIM,A0PT, TITLE H 20 C* IBM REAL*8 XDEL,YDEL,S, AREA,SUMT, RHO.PARAM, TEST,TOL, PINT,HMIN,PYR H 30 LEVEL 2,N0DEID,NPCELL,NPOLD, LIMB 0,IX0BS,IY0BS,THCK PERM,TMWL N1562 LEVEL 2,TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT, TITLE N1563 LEVEL 2,TMOBS N1564 COMMON /PRMI/ NTIM,NPMP,NPNT,NITP.N,NX,NY,NP,NREC, INT,NNX,NNY,NUM0 H 40 IBS,NMOV,IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N H 50 2PNCHV,NPDELC ICHK H 61 COMMON /PRMC/ N0DEID(42,42).NPCELL(42,42),NP0LD(42,42), LIMB 0(500), H 65 IIX0BS(10),IY0BS(10) H 66 2,NDECAY,NSORB N1570 COP 910N /HEDA/ THCK(42,42), PERM (42,42),TMWL(10,50),TM0BS(50),ANFCTR H 90 COMMON /HEDB/ TMP.X(42,42,2),VPRM(42,42) HI(42,42),HR(42,42),HC(42, H 100 l 142) HK(42,42),WT(42,42), REC (42,42),RECH(42,42). TIM (100),A0PT(42) T H 110 l 2ITLE(10),XDEL,YDEL,S AREA,SUMT,RH0,PARAM, TEST,TOL, PINT,HMIN,PYR H 120 COMMON / BALM / TOTLQ,TOTLQI TPIN,TP0UT H 126 DIMENSION IH(42) H 140 C *************************************************************** H 150 TIMD=SUMT/86400. H 160 3 TIMY=SUMT/(86400.0*365.25) H 170 C ---PRINT HEAD VALUES--- H 180 WRITE 6,120 H 190 WRITE 6,130 N H 200 WRITE 6,140 SUMT H 210 WRITE 6,150 TIMD H 220 WRITE 6,160 TIMY H 230 WRITE 6,170 H 240 D0 10 IY=1,NY H 250 10 WRITE (6,180)(HK(IX,IY),IX=1,NX) H 260 IF (N.EQ.0) GO TO 110 H 270 C *************************************************************** H 280 C ---PRINT HEAD MAP--- H 290 WRITE (6,120 H 300 WRITE (6,130 N H 310 WRITE (6,140 SUMT H 320 WRITE (6,150 TIMD H 330 WRITE (6,160)TIMY H 340 WRITE (6,170) H 350 DO 30 IY=1,NY H 360 DO 20 IX=1,NX H 370 20 IH(IX)=HK(IX,IY)+0.5 H 380 1

NUREG-1101, Vol. 3 C-47

30 WRITE (6,190)(IH(ID),ID=1,NX) H 390 C *************************************************************** H 400 C ---COMPUTE WATER BALANCE AND DRAWDOWN--- H 410 H 420 QSTR=0.0 PUMP =0.0 H 430 H 432 PQIN=0.0 H 434 PQ0UT=0.0 TPUM=0.0 H 440

' H 450 QIN=0.0 H 460 Q0VT=0.0 H 470 QNET=0.0 H 480 DELQ=0.0 PCTERR=0.0 H 500 WRITE (6,290) H 510 C

H 520 DO 80 IY=1,NY H 530 _

DO 70 IX=1,NX H 540 H 550 IH(IX)=0.0 H 560 IF (THCK(IX,1Y).EQ.0.0) GO TO 70 IF (REC (IX,IY).GT.0.0) GO TO 32 H 562 PQIN=PQIN+ REC (IX,1Y) H 564 GO TO 34 H 566 32PQ0VT=PQ0UT+ REC (IX,IY) H 568 34 IF (RECH(IX,1Y).GT.0.0) GO TO 36 H 572 PQIN=PQIN+RECH(IX,1Y)* AREA H 574 GO TO 38 H 576 36PQ0VT=PQ0VT+RECH(IX,1Y)* AREA H 578 38 IF (VPRM(IX,IY).EQ.0.0) GO TO 60 H 582 DELQ=VPRM(IX,IY)* AREA *(WT(IX,IY)-HK(IX,1Y)) H 590 H 600 IF (DELQ.GT.0.0) GO TO 40 H 610 000T=Q0VT+DELQ GO TO 50 H 620 H 630 40 QIN=QIN+DELQ H 640 50 QNET=QNET+DELQ 60DDRW=HI(IX,IY)-HK(IX,1Y) H 650 IH(IX)=0DRW+0.5 H 660 H 670 QSTR=QSTR+DDRW* AREA *S 70 CONTINUE H 680 C ---PRINT DRAWDOWN MAP--- H 690 WRITE (6,300)(IH(IX)IX=1,NX) H 700 80 CONTINUE H 710 TPUM = PQIN + PQ0VT H 716 PUMP = TPIN + TP0UT H 721 NUREG-1101, Vol. 3 C-48

TOTLQN=T0TLQ+T0TLQI H 745 SRCS=QSTR-TPIN+TOTLQI H 755 SINKS =TP0UT-TOTLQ H 765 ERRMB=SRCS-SINKS H 775 DEN 0M=(SRCS+ SINKS)*0.5 H 785 IF (DENOM.EQ.0.0) GO TO 100 H 795 PCTERR=ERRMB*100.0/DENOM H 805 C ---PRINT MASS BALANCE DATA FOR FLOW MODEL--- H 830 100 WRITE 6,240 H 840 WRITE 6,211 TPIN H 842 WRITE 6,212 TP0UT H 844 WRITE (6,250 PUMP H 850 WRITE (6,230) QSTR H 860 WRITE (6,202) T0TLQI H 862 WRITE (6,203)TOTLQ H 864 WRITE (6,260 T0TLQN H 866 i WRITE (6,270 ERRMB H 880 WRITE (6,280 PCTERR l H 883 WRITE (6,201 H 886 WRITE (6,202)QIN H 889 WRITE (6,203) 00VT H 892 WRITE (6,204) QNET H 895 WRITE (6,211) PQIN H 898 WRITE (6,212)PQ0VT H 901 WRITE (6,210) TPUM H 910 C *************************************************************** H 940 110 RETURN H 950 C *************************************************************** H 960 C

C H 970 C

H 980 H 990 120 FORMAT (1H1,23HHEAD DISTRIBUTION - R0W) H1000 130 FORMAT 1X.23HNUMBER OF TIME STEPS = ,115) H1010 140 FORMAT 8X,16HTIME(SECONDS) = .1P1E12.5) H1020 150 FORMAT 8X,16HTIME(DAYS) = ,1 PIE 12.5) H1030 160 FOR%T (8X,16HTIME(YEARS) = ,1P1E12.5) H1040 170 FORMAT 1H ) H1050 180 FORMAT IHO,10F12.7) H1055 190 FORMAT 1HO,20I4) H1070 201 FORMAT 1H0,2X,33HRATEMASSBALANCE--(INC.F.S.)//) H1073

=

202 FORMAT (4X.29HLEAKAGE INTO AQUIFER 1 PIE 12.5) H1076 203 FORMAT (4X,29HLEAKAGE OUT OF AQUIFER =

1 PIE 12.5) H1083 204 FORMAT (4X 29HNET LEAKAGE (QNET) = ,1P1E12.5) H1086 NUREG-1101, Vol. 3 C-49

.. . - - ~. -_. . - . _ _ _ . . . .-

c 1

210 FORMAT 4X,29HNET WITHDRAWAL (TPUM) = .1P1E12.5 H1093 211 FORMAT 4X 29HRECHARGE AND INJECTION = ,1P1E12.5 H1096 212 FORMAT 4X,29HPUMPAGE AND E-T WITHDRAWAL = ,1P1E12.5 H1103 230 FORMAT 4X,29HWATER RELEASE FROM STORAGE = .1 PIE 12.5) H1120 240 FORMAT 1HO,2X,38HCUMULATIVE MASS BALANCE -- (IN FT**3) //) H1125 250 FORMAT (4X,29HCUMULATIVE NET PUMPAGE = ,1P1E12.5) H1140 260 FORMAT 4X.29HCUMULATIVE NET LEAKAGE = ,1P1E12.5) H1150 270 FORMAT 1HO,7X,25HMASS BALANCE RESIDUAL = ,1P1E12.5) H1160 280 FORMAT 1H ,7X.25HERROR (AS PERCENT) = 1P1E12.5/) H1170 290 FORMAT 1H1,8HDRAWDOWN) H1180 300 FORMAT 3H 20!S) H1190 END H1200-SUBROUTINE CHMOT I 10 C* IBM REAL*8 TMRX,VPRM,HI,HR,HC.HK WT, REC,RECH, TIM.A0PT, TITLE I 20 C* IBM REAL*8 XDEL,YDEL S. AREA,5UMT. RHO,PARAM. TEST,TOL, PINT,HMIN.PYR I 30 4

LEVEL 2,NODEID,NPCELL,NPOLD, LIMBO,IX0BS.IYOBS,THCK, PERM.TMWL,TMOBS N1572 LEVEL 2,TMRX,VPRF,HI,HR HC,HK,WT, REC,RECH TIM,A0PT, TITLE,PART. CONC N1573

LEVEL 2,TMCN,VX,VY,CONINT.CNRECH N1574 COMON / PRMI / NT IM , NPMP , NPNT .N I TP .N , NX , NY , NP .NREC , I NT , NNX , NNY , NUM0 I 40 IBS,NMOV,IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N I 50

! 2PNCHV,NPDELC.ICHK I 61 COMON /PRMC/ N00EID(42,42),NPCELL(42,42),NPOLD(42,42), LIMB 0(500), I 65 4

IIX0BS(10),IYOBS(10) I 66 2,NDECAY NSORB N1580 COMON/HEDA/THCK(42,42), PERM (42,42),TMWL(10,50),TMOBS(50),ANFCTR I 90 COMON /HEDB/ TMRX(42,42,2),VPRM(42,42),HI(42,42),HR(42,42),HC(42 I 100 l 142),HK(42,42),WT(42,42), REC (42,42),RECH(42,42). TIM (100),A0PT(42).T I 110 21TLE(10) XDEL,YDEL,S. AREA,SUMT, RHO PARAM, TEST,TOL, PINT,HMIN,PYR I 120 COMON /CHMA/ PART(3,9000), CONC (42,42) TMCN(10,50),VX(42,42),VY(42 I 130 1,42).CONINN42,42) CNRECH(42,42),POROS,SUMTCH, BETA TIMV, STORM STOR I 140 2MI CMSIN,CM OUT,FLMIN,FLMOT,SUMIO.CELDIS DLTRAT CSTORM I 150 3,DCYLAM BLKDEN,SRBRAT.SRBSAT,SRBALF,VOLDCY,VOLSRB,SRBDCY N1590 DIMENSION IC(42)

I 160 1

C *************************************************************** I 170 TMFY=86400.0*365.25 1 180 TMYR=SUMT/TMFY I 190 I 200 4

TCHD=SUMTCH/86400.0 TCHYR=SUMTCH/TMFY I 210 ERR 1=0.0 1 212 ERR 3=0.0 I 214 IF (IPRNT.GT.0) GO TO 100 I 220 C *************************************************************** I 230 C ---PRINT CONCENTRATIONS--- I 240 i

NUREG-1101, Vol. 3 C-50

WRITE (6,160)

WRITE (6,170) N I 250 I 260 IF (N.GT.0) WRITE (6,180) TIM (N) I 270 WRITE (6,190) SUMT WRITE (6,450) SUMTCH I 280 WRITE (6,200) TCHD I 290 WRITE (6,210 TMYR I 300 WRITE (6,460 TCHYR I 310 WRITE (6,380 IMOV I 320 WRITE (6,220 I 330 00 20 IY=1,NY 1 340 00 10 IX=1,NX I 350 l 10 IC(IX)= CONC (IX,IY)+0.5 I 360 20 WRITE (6,240) (IC(IX),IX=1,NX) I 370 I 380 C *************************************************************** I 390 IF(N.EQ.0)GOTO150 I 400 IF (NPDELC.EQ.0) GO TO 50 I 410 C

C I 420

---PRINT CHANGES IN CONCENTRATION--- I 420 WRITE (6,230)

WRITE (6,170)N I 440 WRITE (6,180) TIM (N) I 450 WRITE (6,190 SUMT I 460 WRITE (6,450 SUMICH I 470 WRITE (6,200 TCHD I 480 WRITE (6,210 TMYR I 490 WRITE (6,460) TCH)R I 500 WRITE (6,380)IMOV I 510 WRITE (6,220) I 520 00 40 IY=1,NY I 530 00 30 IX=1,NX I 540 CNG= CONC (IX,IY)-CONINT(IX,IY) I 550 30 IC(IX)=CNG I 560 40 WRITE (6,240) (IC(IX),IX=1,NX) I 570 I 580 C

C

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • I 590

---PRINT MASS BALANCE DATA FOR SOLUTE--- I 600 50 RESID=SUMIO-CSTORM + VOLDCY - VOLSRB + SRBDCY N1600 SUMIN=FLMIN-CMSIN I 615 IF (SUMIN.EQ.0.0) GO TO 60 I 625 ERR 1=RESID*100.0/SUMIN 60 IF (STORMI.EQ 0.0) GO TO 70 I 635 I 650 ERR 3=-100.0*RESID/(STORMI-SUMIO) I 660 70 WRITE (6,220)

I 670 i

NUREG-1101, Vol. 3 C-51

WRITE (6,250) I 680 WRITE (6,220) I 690 WRITE (6,260) FLMIN I 700 WRITE (6,270) FLMOT I 710 RECIN=-CMSIN I 720 REC 0UT=-CMSOUT I 730 WRITE (6,290)RECIN I 740 WRITE (6,280) RECOUT I 750 I 760 WRITE (6.300) SUMIO I 770 WRITE (6,310) STORMI I 780 WRITE (6.320) STORM I 790 WRITE (6.330) CSTORM N1610 IF (NDECAY.GT.0) WRITE (6,333) VOLDCY IF (NSORB.GT.0) WRITE (6,335) VOLSRB N1620

' IF (NS0RB.GT.0.AND.NDECAY.GT.0) WRITE (6,336) SRBDCY N1630 I 810 WRITE (6.340) I 820 WRITE (6,350)RESID WRITE (6,360) ERR 1 I 830 I 840 80 IF (STORMI.EQ.0.0) GO TO 90 I 850 WRITE (6,370)

WRITE (6,360) ERR 3 I 860

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • I 870 C

---PRINT HYDR 0 GRAPHS AFTER 50 STEPS OR END OF SIMULATION---

I 880 C

90 IF MOD (IMOV,50).EQ.0.AND.S.EQ.0.0) GO TO 100 I 890 IF M0D(N,50).EQ.0.AND.S.GT.0.0) GO TO 100 I 900 I 905 IF S.EQ.0.0.AND.N.LT.NTIM.AND. INT.GT.0) GO TO 100 I 910 GO TO 150 - I 920 100 WRITE (6.390) TITLE I 930 IF (NUM0BS.LE.0) GO TO 150 I 940 WRITE (6,400) INT IF(S.GT.0.0) WRITE (6,410) I 950 IF (S.EQ.0.0) WRITE (6,420) I 960

---TABULATE HYDR 0 GRAPH DATA- -

I 970 C

1 980 M0Z=0 I 990 IF (S.GT.0.0) GO TO 110 11000 NT0=NM0V IF (NM0V.GT.50) NT0= MOD (IM0V,50) Il010 11020 GO TO 120 11030 110 NT0=NTIM 11040 IF (NTIM.GT.50) NT0= MOD (N,50) 11050 120 IF (NTO.EQ.0) NT0=50 11060 00 140 J=1,NUM0BS 11070 TMYR=0.0 NUREG-1101, Vol. 3 C-52

WRITE (6,430) J,IX0BS(J),IY0BS(J) 11080 WRITE (6,440)M0Z,WT(IX0BS(J),IY0BS(J)),CONINT(IX0BS(J),IY0BS(J)), 11090

.1TMYR I1100 00 130 M=1,NTO I1110 TMYR=TM0BS(M)/TMFY I1120 130 WRITE (6,440) M,TMWL(J,M),TMCN(J,M),TMYR I1130 140 CONTINUE 11140 C **************************************************************** 11150 150 IPRhT = 0 11155 RETURN 11165 C **************************************************************** I1170 C

11180 C

11190 C

11200 160 FORMAT (1H1,13HCONCENTRATION/) 11210 170 FORMAT (1X,23HNUMBER OF TIME STEPS = ,115) 11220 180 FORMAT (8X,16HDELTA T = ,1 PIE 12.5) 11230 190 FORMAT (8X,16HTIME(SECONDS) = ,1 PIE 12.5) 11240 200 FORMAT (3X,21HCHEM. TIME (DAYS) = ,1P1E12.5) 11250 210 FORMAT (8X,16HTIME(YEARS) = ,1P1E12.5) 11260 220 FORMAT (IH ) 11270 230 FORMAT (1H1,23HCHANGEINCONCENTRATION/) 11280 240 FORMAT (1H0,20I5) 11290 250 FORMAT (IH ,21HCHEMICAL MASS BALANCE) 11300 260 FORMAT (8X 25HMASS IN BOUNDARIES = ,1P1E12.5) 11310 270 FORMAT (8X,25HMASS OUT BOUNDARIES = ,1 PIE 12.5) I1320 280 FORMAT (8X 25HMASS PUMPED OUT = .1 PIE 12.5) 11330 290 FORMAT (8X.25HMASS PUMPED IN = ,1 PIE 12.5) 11340 300 FORMAT (8X,25HINFLOW MINUS OUTFLOW = ,1P1E12.5) 11350 310 FORMAT (8X.25HINITIAL MASS STORED = ,1 PIE 12.5) 11360 320 FORMAT (8X,25HPRESENT MASS STORED

= .1 PIE 12.5) 11370 330 FORMAT (8X,25HCHANGE MASS STORF0 l = 1P1E12.5) 11380 333 FORMAT (8X.25HDECAY OF SOLUTE MASS = ,1 PIE 12.5) N1640 335 FORMAT (8X,25HSORBTION STORAGE (S) = ,1P1E12.5) N1650 336 FORMAT (8X,25HS0RBTION DECAY (S) = ,1 PIE 12.5) N1660 340 FORMAT (IH ,5X,53HCOMPARE RESIDUAL WITH NET FLUX AND MASS ACCUMULA 11390 1 TION:) 11400 350 FORMAT (8X,25HMASS BALANCE RESIDUAL = ,1P1E12.5) 11410

=

360 FORMAT (8X 25HERROR (AS PERCENT) 1P1E12.5) 11420 370 FORMAT (1H ,5X,55HCOMPARE INITIAL MASS STORED WITH CHANGE IN MASS 11430 1 STORED:) 11440 380 FORMAT (1X.23H NO. MOVES COMPLETED = .115) 11450 390 FORMAT (1H1,10A8//)

11460 NUREG-1101, Vol. 3 C-53 i

400 FORMAT (1H'0,5X,65HTIME VERSUS HEAD AND CONCENTRATION AT SELECTED 0 11470 IBSERVATION POINTS //15X,19HPUMPING PERIOD NO. ,14////) 11480 410 FORMAT (1HO,16X,19HTRANSIENT SOLUTION ////) 11490 420 FORMAT (1H0,15X,21HSTEADY-STATE SOLUTION ////) 11500 430 FORMAT (1HO,20X,22H0BS.WELL NO. X Y,17X,1HN,6X,40HHEAD (FT) 11510 1 CONC (MG/L) TIME (YEARS)//24X,13,9X,12,3X,12//) 11520 440 FORMAT 1H 58X,12,6X,F7.1,8X,F7.2,8X,F7.3) 11531 450 FORMAT 1H ,2X,21HCHEM. TIME (SECONDS) = ,1P1E12.5) 11540 460 FORMAT 1H ,2X,21HCHEM. TIME (YEARS) = ,1P1E12.5) 11550 END 11560-SUBROUTINE VELNEW (CN0LD) NJ 10 C* IBM REAL*8 TMRX,VPRM,HI,HR.HC,HK,WT, REC,RECH, TIM,A0PT, TITLE NJ 20 C* IBM REAL*8 XDEL,YDEL,S AREA,50MT, RHO,PARAM, TEST ,TOL, PINT,HMIN,PYR NJ 30 C* IBM REAL*8 RATE,SLEAK,DIV NJ 40 LEVEL 2,N0DEID,NPCELL,NPOLD, LIMB 0,IX0BS,1Y0BS,THCK, PERM,TMWL,TM0BS NJ 42 LEVEL 2,TMRX,VPRM,HI,HR,HC,HK,WT, REC,RECH, TIM,A0PT, TITLE,PART, CONC NJ 43 LEVEL 2,TMCN,VX,VY,CONINT,CNRECH,SUMC,VXBDY,VYBDY, DISP NJ 44 COMMON /PRMI/ NTIM,NPMP,NPNT,NITP,N,NX,NY,NP,NREC, INT,NNX,NNY,NUM0 NJ 50 IBS,NMOV,IMOV,NPMAX,ITMAX,NZCRIT,IPRNT,NPTPND,NPNTMV,NPNTVL,NPNTD,N NJ 60 2PNCHV,NPDELC,1CHK NJ 70 COMMON /PRMC/ N0DEID(42,42),NPCELL(42,42),NPOLD(42,42), LIMB 0(500), NJ 80 IIX0BS(10),1Y0BS(10) NJ 90 2 NDECAY,NSORB NJ 100 COMMON /HEDA/THCK(42,42), PERM (42,42).TMWL(10,50)TMOBS(50),ANFCTR NJ 110 COMMON /HEDB/ TMRX(42,42,2),VPRM(42,42),HI(42,42),HR(42,42),HC(42, NJ 120 142),HK(42,42),WT(42,42), REC (42,42),RECH(42,42), TIM (100),A0PT(42),T NJ 130 2ITLE(10),XDEL,YDEL.S. AREA,SUMT, RHO,PARAM, TEST,TOL, PINT,HMIN,PYR NJ 140 COMMON /XINV/ DXINV,DYINV,ARINV,PORINV NJ 150 COMMON /CHMA/ PART(3,9000), CONC (42,42),TMCN(10,50),VX(42,42),VY(42 NJ 160 1,42),CONINT(42,42).CNRECH(42,42),POROS,SUMTCH, BETA,TIMV, STORM.STOR NJ 170 2MI,CMSIN CMS 0VT,FLMIN,FLMOT,SUMIO.CELDIS,DLTRAT,CSTORM NJ 180 3,0CYLAM,BLKDEN,SRBRAT,SRBSAT,SRBALF,VOLDCY,VOLSRB,SRBDCY NJ 190 COMMON /CHMC/ SUMC(42,42),VXBDY(42,42),VYBDY(42,42) NJ 200 COMMON /DIFUS/ DISP (42,42,4) NJ 210 DIMENSION CNOLD(42,42) NJ 220 C *************************************************************** NJ 230 C -------------UPDATE VELOCITIES AND STORE------------ NJ 240 VMAX = 0.0 NJ 250~

TDIV = 0.0 NJ 260 TDIS = 0.0 NJ 270 NJ 280 IXQMAX = 0 NJ 290 IYQMAX = 0

' NJ 300 IF (IMOV.LT.2) ELPTIM = 0.0 NUREG-1101, Vol. 3 C-54

ELPTIM = ELPTIM + TIMV NJ 310 TIMREM = TIM (N) - ELPTIM NJ 320 l RATIOT = TIMREM/ TIM (N) NJ 330 ,

IF (RATIOT.GT.1.0E-07) GO TO 10 NJ 340 l NM0V = IMOV NJ 350 l RETURN NJ 360 10 00 50 IX=1,NX NJ 370 DO 50 IY=1,NY NJ 380 l D0 20 IZ=1,4 NJ 390  !

20 DISP (IX,IY,IZ)=0.0 NJ 400 IF(THCK(IX,IY).EQ.0.0)GOTO50 NJ 410 DISTN = 0.0 NJ 420 DISTO = 0.0 NJ 430 IF (NSORB.GT.2) GO TO 30 NJ 440 C*** ******************LANGMOIR SORBTION****************************** NJ 450 DISTN=BLKDEN*SRBRAT*SRBSAT/(1.0+SRBRAT* CONC (IX,IY))**2.0 NJ 460 DISTO = BLKDEN*SRBRAT*SRBSAT/(1.0+SRBRAT*CN0LD(IX,1Y))**2.0 NJ 470 GO TO 40 -

NJ 480 30 IF (SRBALF.EQ.0.0) GO TO 40 NJ 490 C*** ******************FREUNDLICH S0RBTION**************************** NJ 500 DIST = BLKDEN*SRBRAT*SRBALF NJ 510 LOGCON = -23.0 NJ 520 IF (CONC (IX,IY).GT.1.0E-10) LOGCON = ALOG(CONC (IX,1Y)) NJ 530 SRBEXP = (SRBALF-1.0)*LOGCON NJ 540 IF (SRBEXP.GT.23.0) SRBEXP = 23.0 NJ 550 DISTN = DIST*EXP(SRBEXP) NJ 560 LOGCON = -23.0 NJ 570 IF (CNOLD(IX,1Y).GT.I.0E-10) LOGCON = ALOG(CN0LD(IX,1Y)) NJ 580 SRBEXP = (SRBALF-1.0)*LOGCON NJ 590 IF (SRBEXP.GT.23.0) SRBEXP = 23.0 NJ 610 DISTO = DIST*EXP(SRBEXP) NJ 620 40 RETARD = (1.0+ DISTO)/(1.0+DISTN) NJ 630 C

C NJ 640

---UPDATE VELOCITIES AT N0 DES--- NJ 650 C ---X-DIRECTION--- NJ 660 VX(IX,IY) = VX(IX,IY)* RETARD NJ 670 VX2 = VX(IX,IY)*VX(IX,IY) NJ 680 IF (VX2.GT.VMAX) VMAX = VX2 NJ 690 C ---Y-DIRECTION--- t'l 700 VY(IX,IY) = VY(IX,1Y)* RETARD I,J 710 VY2 =VY(IX,IY)*VY(IX,IY) NJ 720 IF (VY2.GT.VMAX) VMAX = VY2 NJ 730 C ------VELOCITIES AT CELL B0UNDARIES------ NJ 740 NUREG-1101, Vol. 3 C-55

VXBDY(IX,IY) = VXBDY(IX,1Y)* RETARD NJ 750 VX2 = VXBDY(IX,IY)*VXBDY(IX,IY) NJ 760 IF (VX2.GT.VMAX) VMAX = VX2 NJ 770 VYBDY(IX,IY) = VYBDY(IX,IY)* RETARD NJ 780 VY2 = VYBDY(IX,1Y)*VYBDY(IX,1Y) NJ 790 IF (VY2.GT.VMAX) VMAX = VY2 NJ 800 RATE = -REC (IX,1Y)/ AREA NJ 810 SLEAK = (WT(IX,1Y)-HK(IX,IY))*VPRM(IX,1Y) NJ 820 DIV = RATE +SLEAK-RECH(IX,1Y) NJ 830 IF (DIV.LE.0.0) GO TO 50 NJ 840 RTDIV = DIV/(POR0S*THCK(IX,IY)) NJ 850 IF (RTDIV.LT.TDIV) GO TO 50 NJ 860 TDIV = RTDIV NJ 870 IXQMAX = IX NJ 880 IYQMAX = IY NJ 890 50 CONTINUE NJ 900 NJ 910 IF (BETA.EQ.0.0) GO TO 100 C **************************************************************** NJ 920 C ---COMPUTE DISPERSION C0EFFICIENTS--- NJ 930 ALPHA = BETA NJ 940 ALNG= ALPHA NJ 950 TRAN=DLTRAT* ALPHA NJ 960 XX2=XDEL*XDEL NJ 970 YY2=YDEL*YDEL NJ 980 XY2=4.0*XDEL*YDEL NJ 990 DO 70 IX=2,NNX NJ1000 DO 70 IY=2,NNY NJ1010 IF (THCK(IX,IY).EQ.0.0) GO TO 70 NJ1020 VXE=VXBDY(IX,1Y) NJ1030 VYS VYBGY(IX,1Y) NJ1040 IF (THCK(IX+1,IY).EQ.0.0) GO TO 60 NJ1050 C ---F0RWARD COEFFICIENTS: X-DIRECTION--- NJ1060 VYE=(VYBDY(IX,1Y-1)+VYBDY(IX+1,IY-1)+VYS+VYBDY(IX+1,IY))/4.0 NJ1070 VXE2=VXE*VXE NJ1080 VYE2=VYE*VYE NJ1090 NJ1100 VMGE=SQRT(VXE2+VYE2)

NJ1110 IF (VMGE.LT.1.0E-20) GO TO 60 DALN=ALNG*VMGE NJ1120 DTRN=TRAN*VMGE NJ1130 VMGE2=VMGE*VMGE NJ1140

---XX COEFFICIENT--- NJ1150 C

DISP (IX,IY,1)=(DALN*VXE2+DTRN*VYE2)/(VMGE2*XX2) NJ1160

---XY COEFFICIENT--- NJ1170 C

NUREG-1101, Vol. 3 C-56

t t

l DISP (IX,IY,3)=(DALN-DTRN)*VXE*VYE/(VMGE2*XY2) NJ1180 C ---FORWARD C0EFFICIENTS: Y-DIRECTION--- NJ1190 60 IF (THCK(IX,IY+1).EQ 0.'0) GO TO 70- NJ1200 VXS=(VXBDY(IX-1,1Y)+VXE+VXBDY(IX-1,1Y+1)+VXBDY(IX ,'IY+1))/4.0 - NJ1210 VYS2=VYS*VYS NJ1220 VXS2=VXS*VXS- NJ1230 VMGS=SQRT(VXS2+VYS2) NJ1240 IF (VMGS.LT.1.0E-20) GO TO 70 NJ1250 DALN=ALNG*VMGS NJ1260 DTRN=TRAN*VMGS NJ1270 VMGS2=VMGS*VMGS NJ1280 C ---YY C0 EFFICIENT--- NJ1290 DISP (IX,IY,2)=(DALN*VYS2+DTRN*VXS2)/(VMGS2*YY2) NJ1300 C ---YX COEFFICIENT-- -.. 'NJ1310 DISP (IX,IY,4)=(DALh-DTRN)*VXS*VYS/(VMGS2*XY2) NJ1320 70 CONTINUE NJ1330 C **************************************************************** NJ1340 C ---ADJUST CROSS-PRODUCT TERMS FOR ZER0 THICKNESS--- NJ1350 DO 80 IX=2,NNX NJ1360 DO 80 IY=2,NNY NJ1370 IF(THCK(IX,IY+1).EQ.0.0.0R.THCK(IX+1,1Y+1).EQ.0.0.0R.THCK(IX,IY-1 NJ1380 1).EQ.0.0.0R.THCK(IX+1,IY-1).EQ.0.0) DISP (IX,IY,3)=0.0 NJ1390 IF(THCK(IX+1,IY).EQ.0.0.0R.THCK(IX+1,IY+1).EQ.0.0.0R.THCK(IX-1,IY NJ1400 1).EQ.0.0.0R.THCK(IX-1,IY+1).EQ.0.0) DISP (IX,IY,4)=0.0 NJ1410 C **************************************************************** NJ1420 l C ---CHECK FOR STABILITY OF EXPLICIT METHOD--- NJ1430 TDISP = DISP (IX,IY,1)+ DISP (IX,1Y,2) NJ1440 l 80 IF (TDISP.GT.TDIS) TDIS = TDISP NJ1450 C

NJ1460 C ---ADJUST DISP. EQUATION COEFFICIENTS FOR SATURATED TVICKNESS--- NJ1470 00 90 IX=2,NNX NJ1480 D0 90 IY=2,NNY NJ1490 IF (THCK(IX,1Y).EQ.0.0) G0 10 90 -

NJ1500 BAVX=2.0*THCK(IX,IY)*THCK(IX+1,1Y)/(THCK(IX,IY)+THCK(IX+1,IY)) NJ1510 BAVY=2.0*THCK(IX,IY)*THCK(IX,1Y+1)/(THCK(IX,IY)+THCK(IX IY+1)) NJ1520 DISP (IX,1Y,1)= DISP (IX,1Y,1)*BAVX NJ1530 DISP (IX,IY,2)= DISP (IX,IY,2)*BAVY NJ1540 DISP (IX,IY,3)= DISP (IX,1Y,3)*BAVX NJ1550 DISP (IX,1Y,4)= DISP (IX,IY,4)*BAVY NJ1560 90 CONTINUE NJ1570 C **************************************************************** NJ1580 C ---COMPUTE NEXT TIME STEP--- NJ1590 100 VMAX = SQRT(VMAX) NJ1600 NUREG-1101, Vol. 3 C-57 l

l

RTADV = VMAX/(CELDIS*XDEL) NJ1610 TDIS = 2.0*TDIS NJ1620 RTMAX = AMAX1(RTADV,TDIV,TDIS) NJ1630 TIMV = 1.0/RTMAX NJ1640 PMOVRM = TIMREM/TIMV + 1 NJ1650 TIMV = TIMREM/(1.0*NMOVRM) NJ1660 NMOV = IMOV + NMOVRM NJ1670 WRITE (6,6000) NMOVRM,N NJ1680 IF RTMAX.EQ.RTADV) WRITE (6,6010) NJ1690 IF RTHAX.EQ.TDIS) WRITE (6,6020) NJ1700 IF RTMAX.EQ.TDIV) WRITE (6,6030) IX0 MAX,1YQMAX NJ1710 RETURN NJ1720 C **************************************************************** NJ1730 C -------0UTPUT FORMATS-------- NJ1740 C **************************************************************** NJ1750 6000 FORMAT (1H1,10X,38HSTABILITY CRITERIA --- M.O.C. REQUIRES,I3, NJ1760 154HADDITIONAL PARTICLE MOVES TO COMPLETE FLOW TIME STEP =,13) NJ1770 6010 F0kMAT 1H1,10X,42HTHE LIMITING STABILITY CRITERION IS CELDIS) NJ1780 6020 FORMAT 1H1,10X,40HTHE LIMITING STABILITY CRITERION IS BETA) NJ1790 6030 FORMAT 1H1,10X,58HTHE LIMITING STABILITY CRITERION IS MAXIMUM INJ NJ1800 1ECTION RATE //15X,35H MAX. INJECTION OCCURS IN CELL IX = ,13, NJI810 2 6H IY = ,I3) NJ1820 END NJ1830 l

NUREG-1101, Vol. 3 C-58

7 1

., M0CMOD84 SAMPLE INPUT WITH JOB CONTROL USERID,STMFZ,TPO,T777. NAME TELEPHONE ACCOUNT,USERID. ACCOUNT #, ACCOUNT #.

ATTACH',LG0,MOCMOD84 ABS,ID=ZZRNRC MR=1.

LG0,PL=100000.

REWIND, INPUT.

COPYSBF, INPUT.0UTPUT.

• EOR THALF=60 DAYS KD=.1 V=1FT/ DAY N=.1 AX=20 AY=4 B=10 SOURCE =1E9/YR 3APR84 1 1 10 429000 2 7 4 100 2 9 1 40 1 1 0 0 2.5 .001 0.10 20. 0.0 0.0 0.0 25. 25. 0.20 0.3 1.0 1 1 0.1644 2.4 0.1 210 214 220 230 210 .5E-6 3.171E7 240 .5E-6 0.0 01.15741E-3 0 10.0 0 0.0 1 1.0 00000000000000000000 01111111111111111110 00000000000000000000 0

0 0

0

! O-0 0

0 0

0 0

0 0

0 0

0 0

NUREG-1101, Vol. 3 C-59

2 0 ,

0 0

0

0. .

0 0

0 0 '

0 0

0 0

0

-0 0

0 0

0 0

01111111111111111110 0

1- 1.0 0.0 0.0 0 1 1.0

. 00.0 0.0 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10, 10. 10. 10. 10. 10. 10. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 i

1 NUREG-1101, Vol. 3 C-60

. , -_ . . . . -. - -. - .~ . . - . _ . _

s e

1 1  :

0.0 0.0-0.0 0.0 Oro.

0.0 0.0  ;

0.0 0.0 -

0.0 0.0

0.0

! 0.0  !

0.0 0.0 0.0 0.0

0.0 0.0 '

O.0 O.0

j. 0.00.250.250.250.250.250.250.250.250.250.250.250.250.250.250.250.250.250.25 0.0 0.0 0 0.0 ..

• EOR

• E0F l

1 ,

I l l

NUREG-1101, Vol. 3 C-61

E M0CMOD84 SAMPLE OUTPUT (SELECTED) g h

o 10.S.G.S. METHOD-OF-CHARACTERISTICS MODEL FOR SOLUTE TRANSPORT IN GROUND WATER OTHALF=60 DAYS KD=.1 V=1FT/ DAY N=.1 AX=20 AY=4 B=10 SOURCE =1E9/.YR 3APR84

." O INPUT DATA 0 GRID DESCRIPT0RS g

HX = 10 (NUMBEROFCOLUMNS) w NY (NUMBER OF R0WS)

= 42 XDEL (X-DISTANCE IN FEET) = 25.0 YDEL (Y-DISTAhCE IN FEET) = 25.0 0 TIME PARAMETERS

=

NTIM (MAX. K0. OF TIME STEPS) 1 NPMP = 1 (fl0. OF PUMPING PERIODS)

PINT (PUMPING PERIOD IN YEARS) = 2.500 TIMX (TIME INCREMENT MULTIPLIER)

= 0.00 TINIT (INITIAL TlHE STEP IN SEC.) = 0.

O HYDROLOGIC AND CHEMICAL PARAMETERS 7 5 (STORAGE COEFFICIENT) = 0.000000

$ POROS (EFFECTIVE POROSITY) = .10 CETA (CHARACTERISTIC LENGTH)

= 20.0 DLTRAT (RATIO 0F TRANSVERSE TO LONGITUDINAL DISPERSIVITY) = .20 AMFCTR (RATIO 0F T-YY TO T-XX) = 1.000000 0 SPECIES HALF LIFE (YEARS) = 1.644E-01 OR DECAY CONS 1 ANT (1/ SECS) = 1.336E-07 0 ROCK DENSITY (GRM/CM**3) = 2.400E+00B0LK DEhSITY/ POROSITY = 2.160E+01 0 "* LINEAR SORBTION*** DISTRIBUTION CONSTANT (KD) 1.000E-01 0 EXECUTION PARAMETERS NITP (h0. OF ITERATION PARAMETERS) = 7 TUL (CONVERGENCE CRITERIA - ADIP) = .0010 ITMAX (MAX.NO.0F ITERATIONS - ADIP) = 100 CELUIS (hAX. CELL DISTANCE PER MOVE

E

=

h 0F PARTICLES - M.O.C.) = .300 0

o NPPAX (MAX. NO. 0F PARTICLES) = 9000

" NPTPND (NO. PARTICLES PER N0DE) = 9.

O PROGRAM OPTIONS NPt4T (TIME STEP INTERVAL FOR

= 2 W COMPLETE PRINT 0UT) flPNTMV (MOVE IhTERVAL FOR CHEM.

= 40 CONCENTRATION PRINT 0UT) hPNTVL (PRINT OPTION-VELOCITY 0=N0; 1=FIRST TIME STEP; 2=ALL TIME STEPS) = 1 NPMTD (PRINT OPTION-DISP.00EF.

0=N0; I=FIRST TIME STEP; 2=ALL TIME STEPS) = 1 NUMOBS (NO. OF OBSERVATION WELLS FOR HYDR 0 GRAPH PRINT 0UT) = 4 NREC (NO. OF PUMPING WELLS) = 2 n NC0 DES (FOR N0DE IDENT.) = 1 4

w NPNCHV (PUNCH VELOCITIES) = 0 NPDELC (PRINT OPT.-CONC. CHANGE) = 0 1 STEADY-STATE FLOW 0 TIME INTERVAL (IN SEC) FOR SOLUTE-TRANSPORT SIMULATION = 7.88940E+07 LOCATION OF OBSERVATION WELLS NO. X Y 1 2 10 2 2 14 3 2 20 4 2 30 0 LOCATION OF PUMPlf!G WELLS

A-m E

= X Y RATE (IN CFS) CONC.

g) w ********

E$ 2 10 .0000 2 40 .0000 0.00

!" 0 AREA 0F ONE CELL = 6.2500E+02

$; O X-Y SPACING:

- 25.000 us 25.000 ITRANSMISSIVITY MAP (FT*FT/SEC) O. O.

O. O. O. O. O.

0. O. O.

0.

! 0. 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 0.

! 0, 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 0.

! 0. 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 l . . . . .

n g 0. 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 0.0.

O. 1.16E-03 '1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 0.

O. 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 1.16E-03 O. O.

O. O. O. O. O. O. O.

O.

IAQU1FER THICKNESS (FT) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 0.0 0.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 0.0 0.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 0.0 N N N N N M M . M M 0.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 0.0 0.0 10.0 10.0 10.0 10.0'10.0 10.0 10.0 10.0 0.0

0.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

'l

E E IDiff uSE RECHARGE AND DISCHAl(GE (FT/SEC)

? 0. O. O. O. O. O. O. O. O. O..

t 0. O. O. O. O. O. O. O. o. O.

p 0. O. O. O. O. O. O. O. O. O.

S o.

0. O. O. O. o. O. O. O. O.

" 0. O. O. O. O. O. O. O. O. O.

O. O. O. O. O. O. O. O. O. O.

l l

IPERMEABILTY MAP (FT/SEC)

o. O. O. O. O. O. O. O. O. O.

O. 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 0.

O. 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 0.

O. 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 0, i ? - . . . . . . . . .

0. 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 0.

O. 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E 1.16E-04 1.16E-04 0.

O. 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 1.16E-04 0.

O. O. O. O. O. O. O. O. O. O.

0 i N0. OF FINITE-OlFFERENCE CELLS IN AQUIFER = 320

=

AREA 0F AQUIFER IN MODEL 2.00000E+05 SQ. FT.

i NZCRIT (MAX. NO. OF CELLS THAT CAN BE VOID OF

PARTICLES; IF EXCEEDED, PARTICLES ARE REGENERATED) = 6 IN0DE IDENTIFICATION MAP

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NUREG-1101, Vol. 3 C-67

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' T' 0 0.0000000' 7.5005108 7.5004853 7.5004603 7.5004392 7.5004191 7.5004001 7.5003865 7.5003829 0.0000000 O 0.0000000 7.2504532 7.2504365 7.2504206 7.2504063 7.2503915 7.2503772 7.2503683 7.2503691 0.0000000 N 0 0.0000000 7.0004075 7.0003937 7.0003d36 7.0003749 7.0003644 7.0003532 7.0003471 7.0003513 0.0000000 T 0 0.0000000 6.7503869 6.7503762 6.7503711 6.7503672 6.7503599

' 6.7503502- 6.7503445 6.7503491 0.0000000 w 0 0.0000000 6.5003846 6.5003769 6.5003756 6.5003752 6.5003702 6.5003613 6.5003554 6.5003590 0.0000000 0 0.0000000 6.2503861 6.2503801 6.2503809 6.2503826 6.2503792 6.2503711 6.2503654 6.2503688 0.0000000 0 0.0000000 6.0003839 6.0003780 6.0003795 6.0003821 6.0003798 6.0003728 6.0003681 6.0003721 0.0000000-i' 0 0.0000000 5.7503792 5.7503730 5.7503745 5.7503776 5.7503760 5.7503700 5.7503663 5.7503713 0.0000000 O 0.0000000 5.5003769 5.5003707 5.5003723 5.5003756 5.5003746 5.5003692 5 3CJ3661 5.5003716 0.0000000 0 0.0000000 5.2503795 5.2503737 5.2503757 5.2503794 5.2503787 5.2503737- 5.2503707 5.2503759 0

0.0000000 4

0.0000000 5.0003854 5.0003805 5.0003831 5.0003872 5.0003867 5.0003817 5.0003785 5.0003831 0.0000000 0 0.0000000 4.7503908 4.7503865 4.7503897 4.7503941 4.7503938 4.7503868 4.7503853 4.7503893 0.0000000 0 0.0000000 4.5003916 4.5003877 4.5003912 4.5003959 4.5003957 4.5003907 4.5003870 4.5003907 0.0000000 i 0 0.0000000 4.2503855 4.2503818 4.2503853 4.2503901 4.2503900 4.2503851 4.2503815 4.2503852 0 0.0000000 0.0000000 4.0003726 4.0003687 4.0003721 4.0003769 4.0003769 4.0003722 4.0003688 4.0003727 0 0.0000000 0.0000000 3.7503544 3.7503502 3.7503534 3.7503580 3.7503581 3.7503537 3.7503507 3.7503549 0.0000000

?

m 0 0.0000000 3.5003330 3.5003287 3.5003316 3.5003360 3.5003363 3.5003323 3.5003297 3.5003341 0.0000000

• 0 0.0000000 3.2503104 3.2503063 3.2503090 3.2503133 3.2503135 3.2503101 0

3.2503078 3.2503122 0.0000000 0.0000000 3.0002876 3.0002839 3.0002867 3.0002908 3.0002914 3.0002883 0 3.0002863 3.0002904 0.0000000 0.0000000 2.7502642 2.7502612 2.7502640 2.7502682 2.7502691 2.7502664 2.7502647 2.7502684 0.0000000 0 0.0000000 2.5002390 2.5002366 2.5002397 2.5002440 2.5002453 2.5002433 0 0.0000000 2.5002419 2.5002453 0.0000000 2.2502115 2.2502097 2.2502132 2.2502178 2.2502196 2.2502184 2.2502176 0 0.0000000 2.2502208 0.0000000 2.0001816 2.0001809 2.0001850 2.0001902 2.0001929 2.0001927 2.0001926 0 0.0000000 2.0001954 0.0000000

! 1.7501474 1.7501482 1.7501536 1.7501599 1.7501637 1.7501648 0 0.0000000 1.7501653 1.7501679 0.0000000 l

! 1.5001073 1.5001099 1.5001174 1.5001255 1.5001310 1.5001334 0 0.0000000 1.5001350 1.5001380 0.0000000 1.2500632 1.2500699 1.2500814 1.2500924 1.2500999 1.2501039 0 0.0000000 1.2501064 1.2501095 0.0000000 l 1.0000115 1.0000291 1.0000487 1.0000639 1.0000735 1.0000786 0 0.0000000 1.0000813 1.0000834 0.0000000

.7499379 .7499828 .7500154 .7500348 .7500454 .7500507 0 0.0000000 .7500533 .7500549 0.0000000

.4998169 .4999465 .4999916 .5000100 .5000182 .5000218 0 0.0000000 .5000239 .5000257 0.0000000

.2500005 .2500005 .2500005 .2500005 .2500005 .2500005 O 0.0000000 .2500005 .2500005 .2500000 ;

0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 1 HEAD DISTRIBUTION - R0W NUMBER Cf TIME STEPS = 1 TIME (SECONDS) = 7.88940E407

=

TIKE (DAYS) 9.13125E+02 l

_ _ _ . _ . _. . _ , _ . . _ _ _ . . . .- _ _ _ _ _ . . _ . . m . .. _ _ . , . . _ .

.z.

E

$ TIME (YEARS)

= 2.50000E+00

• 0 0 0 0 0 0 0 0 0 0 0

's 0 0 10 10 10 10 10 10 10 10 10

~

0 0 10 10 10 10 10 10 10 ~ 10 0

$ 0 0 10 10 10 10 10 10 10 .10 0

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0 0 5 5 5 5 5 5 5 5 0 0 0 5 5 5 5 5 5 5 5 0 O 0 5 5 5 5 5 5 5 5 0 0 0 5 5 5 5- 5 5 5 5 0 0 0 4 4 4 4 4 4 4 4 0 0 0 4 4 4 4 4 4 4 4 0 0 0 4 4 4' 4 4 4 4 4 0 0 0 4 4 4 4~4 4 4 1 0 0 0 3 3 3 3 3 3 3 3 0 0 0 3 3 3 3 3 3 3 3 0 0 0 3 3 3 3 3 3 3 '3 0 1 0 0 3 3 3 3 3 3 3 3 0 0 0 2 2. 2 2 2- 2 2 2 0 ..

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=

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, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0' O o

P IDRAWDOWN w 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 'O O O 0 -9 -9 -9 -9 -9 -9 -9 -9 0 0 -9 -9 -9 -9 -9 -9 -9 -9 0 0 -8 -8 -8 -8 -8 -8 -8 -8 0 0 -8 -3 -8 -8 -8 -8 -8 -8 0 0 -8 -8 -8 -8 -8 -8 -8 -8 0 0 -8 -8 -8 -8 -8 -8 -8 -8 0 0 -7 -7 -7 -7 -7 -7 -7 -7 0 4 0 -7 -7 -7 -7 -7 -7 -7 0 0 -7 -7 -7 -7 -7 -7 -7 -7 0 0 -7 -7 -7 -7 -7 -7 -7 -7 0 n 0 -6 -6 -6 -6 -6 -6 -6 -6 0 i

0 0 -6 -6 -6 -6 -6 -6 -6 -6 0 O -6 -6 -6 -6 -6 -6 -6 -6 0 0 -6 -6 -6 -6 -6 -6 -6 -6 0 0 -5 -5 -5 -5 -5 -5 -5 -5 0

0 -5 -5 -5 -5 -5 -5 -5 -5 0 0 -5 -5 -5 -5 -5 -5 -5 -5 0' 0 -5 -5 -5 -5 -5 -5 -5 -5 0 0 -4 -4 -4 -4 -4 -4 -4 -4 0 j 0 -4 -4 -4 -4 -4 -4 -4 -4 0 0 -4 -4 -4 -4 -4 -4 -4 -4 0 0 -4 -4 -4 -4 -4 -4 -4 -4 0 i 0 -3 -3 -3 -3 -3 -3 -3 -3 0 0 -3 -3 -3 -3 -3 -3 -3 -3 0 0 -3 -3 -3 -3 -3 -3 -3 -3 0 0 -3 -3 -3 -3 -3 -3 -3 -3 0 0 -2 -2 -2 -2 -2 -2 -2 -2 0 0 -2 -2 -2 -2 -2 -2 -2 -2 0 j 0 -? -2 -2 -2 -2 -2 -2 -2 0 1

0 -2 -2 -2 -2 -2 -2 -2 -2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0

E A

? O -1 -1 -1 -1 -1 -1 -1 -1 0 H 0 -1 -1 -1 -1 -1 -1 -1 -1 0 E$ 0 -1 -1 -1 -1 -1 -1 -1 -1 0

." 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 w 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CUMULATIVE MASS BALANCE -- (IN FT**3)

RECHARGE AND INJECTION = -3.94470E+01 PUMPAGE AND E-T WITHDRAWAL = 3.94470E+01 CUMULATIVE NET PUMPAGE = 0.

WATER RELEASE FROM STORAGE = 0.

LEAKAGE INTO AQUIFER = 1.82642E+05 LEAKAGE OUT OF AQUIFER = -1.82612E+05 CUMULATIVE NET LEAKAGE = 2.99720E+01

? 0 MASS BALANCE RESIDUAL = 2.99720E+01 d ERROR (AS PERCENT) = 1.64081E-02 0 RATE MASS BALANCE -- (IH C.F.S.)

LEAKAGE INTO AQUIFER = 2.31503E-03 LEAKAGE OUT OF AQUIFER = -2.31465E-03 NET 4.EAKAGE (QNET) = 3.79903E-07 RECHARGE AND INJECTION = -5.00000E-07 PUMPAGE AND E-T WITHDRAWAL = 5.00000E-07 NET WITHDRAWAL (TPUM) = 0.

IX VELOLITIES AT N0 DES

0. O. O. O. O. O. O. O. O. O.

E

=

9? 0. 3.777E-16 5.176E-16 1.124E-16 -2.498E-18 9.869E-17 -2.998E-17 -4.618E-16 -3.581E-15 0.

[j 0. 2.040E-10 2.795E-10 6.072E-11 -1.232E-12 5.320E-11 -1.623E-11 -2.493E-10 -1.933E-10 0, o 0. -8.317E-11 -1.331E-10 -4.618E-11 3.256E-11 7.884E-11 1.420E-10 1.985E-10 1.065E-10 0.

I" 0. -9.030E-11 -1.028E-10 .3.158E-11 9.219E-11 9.421E-11 1.304E-10 2.153E-10 1.309E-10 0.

< 0. 1.929E-10 3.275E-10 2.093E-10 1.480E-10 1.377E-10 4.406E-11 -1.459E-10 -1.255E-10 0.

w . . .

0. -3.291E-10 -5.680E-10 -3.809E-10 -2.198E-10 -1.166E-10 -5.766E-11 -3.090E-11 -1.207E-11 0.

! 0. -9.495E-10 -1. 280E-09 -4.652E-10 -1.946E-10 -8.656E-11 -4.177E-11 -2.833E-11 -1.331E-11 0.

0. -1.758E-15 -2.370E-15 -8.615E-16 -3.603E-16 -1.603E-16 -7.735E-17 -5.247E-17 -2.466E-17 0.

l

0. O. O. O. O. O. O. O. O. O.

O ON BOUNDARIES

! 0. O. O. O. O. O. ~0. O. O. O.

O. 7.553E-16 2.798E-16 -5.496E-17 4.997E-17 1.474E-16 -2.074E-16 -7.162E-16 0. O.

O. 4.081E-10 1.508E-10 -2.939E-11 2.693E-11 7.946E-11 -1.119E-10 -3.867E-10 0. O.

O. -1.663E-10 -9.981E-11 7.463E-12 5.765E-11 1.000E-10 1.840E-10 2.131E-10 0. O.

O. -1.806E-10 -2.503E-11 8.819E-11 9.618E-11 9.223E-11 1.686E-10 2.619E-10 0. O.

? 0. 3.859E-10 2.692E-10 1.494E-10 1.466E-10 1.288E-10 -4.068E-11 -2.510E-10 0. O.

U O.

-6.582E-10 -4.778E-10 -2.841E-10 -1.555E-10 -7.765E-11 -3.766E-11 -2.414E-11 0. O.

0. -1.899E-09 -6.610E-10 -2.695E-10 -1.196E-10 -5.349E-11 -3.004E-11 -2.662E-11 0. O.

i 0. -3.517E-15 -1.224E-15 -4.991E-16 -2.215E-16 -9.908E-17 -5.562E-17 -4.932E-17 0.

0. 0.

j O. O. O. O. O. O. O. O. O.

1Y VELOCITIES AT N0 DES 0, 0. O. O. O. O. O. O. O. O.

O. 1.831E-06 1.832E-06 1.832E-06 1.832E-06 1.832E-06 1.832E-06 1.832E-06 1.831E-06 0.

O. 3.662E-06 3.662E-06 3.661E-06 3.661E-06 3.661E-06 3.662E-06 3.662E-06 3.662E 0.

O. 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 0.

O. 3.662E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

E A

o

0. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

/s 0.

O. 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.662E-06 3.662E-06 E' 3.662E-06 0.

>" 0. 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06

~

0. 3.661E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 0.

EE 0. 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 0.

0. 3.664E-06 3.663E-06 3.663E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 0.

u, D. 3.664E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

r, O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

b 0. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

0. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.'663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.664E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.664E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.662E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 1.830E-06 1.831E-06 1.831E-06 1.831E-06 1.831E-06 1.832E-06 1.832E-06 1.832E-06 0.

O. O. O. O. O. O. O. O. O. O.

0 ON BOUNDARIES

Y E

x El 0. O. O. O. O. O. O. O. O. O.

Ja 0. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.661E-06 0.

E$ - 3.660E-06 3.660E-06 3.660E-06 3.660E-06 3.660E-06 3.660E-06 3.661E-06 0.

?" 0. 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 0.

,e 0. 3.663E-06 3.664E-06 3.664E-06 3.664E-06 3.664E-06 3.664E-06 3.664E-06 3.664E-06 0, o 0. 3.662E-06 3.662E-06 3.662E-06 T'

3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 .0 O. 3.661E-06 3.660E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06 3.661E-06

0. 3.661E-06 0.

ca 3.661E-06 3.662E-06 3.662E-06 3.'662E-06 3.662E-06 3.662E-06 3.662E-06 0.

O. 3.660E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.663E-06 3.662E-06 3.664E-06 0.

O. 3.663E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 3.662E-06 0.

! 0. 3.664E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.662E-06 3.662E-06 0.

• i

0. 3.664E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06~ 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

52 0. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06

-4 0. 3.663E-06 0.

i 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-05 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0 O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-C6 3.663E-06 3.663E-06 3.663E-06 I

0. 3.663E-06 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

O. 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.664E-06 3.663E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 O. 3.664E-06 0.

3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 3.663E-06 0.

_. ._ . _. .___ _ _ _ _ . _ _ _ - ._ __ . - ~ _ _ . . _ ._.

i E

M;

'E 0. 3.660E-06 .3.662E-06 3.663E-06 3.663E-06 3.663E-06, 3.663E-06 3.663E 3.663E-06~ 0.

II 0. O. O. O. O. . 0. O. O. 0. O.

0. o. 0. 0. O. o. o. 0.

j3 O. O.

1 UNRETARDED X VELOCITIES '

AT N0 DES O. O. O. O. '0. O. O. O. . 0.- 0.

0.

O. 1.193E-15 1.636E-15 3.553E-16 -7.895E 3.119E-16 -9.474E-17 -1.459E-15 -1.132E O. 6.448E-10 8.831E-10 1.919E-10 -3.893E-12L 1.681E-10 -5.127E-11 -7.878E-10 -6.110E-10 0.

i 0. -2.628E-10 -4.205E-10 -1.459E-10 1.029E-10 2.491E-10 4.488E-10 6.273E-10 3.366E-10 0.

! 0. -2.854E-10 -3.249E-10 9.980E-11. 2.913E-10 2.977E-10 4.122E-10 6.802E-10 4.138E 0. t O. 6.097E-10 1.035E 6.614E-10 4.678E-10 4.352E-10 1.392E-10 -4.609E-10 -3.966E-10 0.

\ . . . . . . . .

0. -1.040E-09 -1.795E-09 -1.204E-09 -6.945E-10 --3.684E-10 -1.822E-10 -9.764E-11 -3.814E-11 0.

b 0.

o' O. -3,000E-09 -4.045E-09 -1.470E-09 -6.148E-10 -2.735E-10 -1.320E-10 -8.953E-11 -4.207E-11

-5.556E-15 -7.490E-15 -2.722E-15. -1.139E-15 -5.066E-16 -2.444E-16 -1.658E-16 -7.792E-17. O.

O.

O. O. O. O. O. O. O. O. O. O.

4 i

1 UNRETARDED Y VELOCITIES

• AT N0 DES i

0, 0. O. .0. O. O. O. O. O. O.

1 0. 5.787E-06 5.788E-06 5.788E-06 5.788E-06 5.788E-06 5.788E 5.788E-06 5.787E-06 0.

! 0. 1.157E-05 '1.157E-05 1.157E-05 1.157E-05 1.157E-05 'I.157E 1.157E 1.157E-05 0.

j 0. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

0. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E 1.157E-05: 1.157E-05 1.157E 0.

O. 1.157E-05 1.157E-05 1.157E-05. 1.158E-05 1.158E-05 1.158E-05: -1.158E-05 1.157E 0. ,

0, 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E 1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E 1.157E-05 1.157E-05' 1.157E-05 1.157E-05 1.157E 1.157E-05 0.

i Z

55 E'

O. 1.157E-05 1.157E-05 -1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E 0.

U$ 0. 1.158E-05 1.157E-05 1.157E 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

1 E3 0. 1.15bE-05 1.158E-05 1.158E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

0. 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.157E-05 0.

gf 0. 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.157E-05 1.157E 1.157E-05 1.157E-05 0.

0. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E 1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E 1.157E-05 1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05. 1.157E-05 0.

I O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

0. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 -0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E 1.157E-05 1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 .1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

j 0. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 l'.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

O. 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 1.157E-05 0.

O. 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.157E-05 0.

! r3 0. 1.158E-05 1.158E-05 1.158E-05 1.168E-05 1.158E-05 1.158E-05 1.158E-05

0. 1.158E-05. O.

Ja 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 0.

N 0.

1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05. 1.158E-05 0.

O. 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E 0.

O. 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E 1.158E-05

0. 1.158E-05 1.158E-05 0.

1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 0.

O. 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 0.

O. 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 G. 1.158E-05 1.158E-05 0.

1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 0.

O. 1.158E-05 1.158E-05 1,158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 O. 1.158E-05 1.158E-05 0. '

1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 0.

O. 1.158E-05 1.158E-05 1.158E-05 1.158E 1.158E-05 1.158E-05 1.158E-05 O. 1.157E-05 1.158E-05 0.

1.157E-05 1.157E-05 1.157E-05 1.158E-05 1.158E-05 1.158E-05 1.158E-05 0.

3 0. .5.783E-06 5.786E-06 5.787E-06 5.787E-06 5.787E-06 5.788E-06 -5.788E-06 O. O.

5.788E-06 0.

, O. O. O. O. O. O.

O. O.

1 STA8ILITY CRITERIA --- M.O.C.

i e

i 1

i _ ._ _ __

~ ~ . . .

A

5) 0 VMAX = 1.71E-09 VMAY = 3.66E-06 9' VMxBD= 2.36E-09 VMYBD= 3.66E-06 E$ TMV (MAX INJ.) = .21597E+07

. TIMV (CELDIS) = 2.04668E+06

<: 0 TIMV = 2.05E+06 NTIMV = 38 NMOV = 39

• a TIM (N) = 7.88940E407 c) TIMEVELO = 2.02292E+06 TIMEDISP = 3.55339E+06 C TIMV = 2.02E+06 NTIMD = 22 NMOV = 39-0 THE LIMITING STABILITY CRITERION IS CELDIS IDISPERSION EQUATION COEFFICIENTS =(D-IJ)*(B)/(GRID FACTOR)

XX COEFFICIENT

0. O. O. O. . O. O. O. O. O. O.

O. 1.2E-07 1.2E-07 1.2E-07 1.2E-07 1.2E-07 1.2E-07 1.2E-07 0. O.

O. 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 0. O.

O. 2.3E-07 2.3E-07 2.3E-07 2.3E-07 ?.3E-07 2.3E-07 2.3E-07 0. O.

'? . . . . . . . . . .

3

0. 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 0. O.

O. 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 2.3E-07 0. O.

O. 1.2E-07 1.2E-07 1.2E-07 1.2E-07 1.2E-07 1.2E-07 1.2E-07 0. O.

O. O. O. O. O. O. O. 0. O. O.

YY COEFFICIENT

0. O. O. O. O. O. O. O. O. O.

0. 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 0.

0. 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 0.

0. 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 0.

O. 1.2E-06 1.2E-06 1.2E-06.1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 0.

O. 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06 1.2E-06'1.2E-06 1.2E-06 0.

0 1.2E-06 1. 2E-06 1.2E-06 -1.2E-06 1.2E 1.2E-06 1.2E-06 1.2E-06 0.

O. O. O. O. O. O. O. O. O. O.

0. O. O. O. O. O. O. O. O. O.

1 i

E

$ XY COEFFICIENT

?

co .O. O. O. O. O. O. O. O. O. O.

0. O. .

O. O. O. 0. O. O. O. O.

0. 2.6E-11 9.7E-12-1.9E-12-1.7E-12 5.1E-12-7.2E-12-2.5E-11 0. O.

<: 0. -1.1E-11-6.4E-12 4.SE-13 3.7E-12 6.4E-12 1.2E-11 1.4E-11 0. O.

O. -1.2E-11-1.6E-17 :).6E-12 6.2E-12 5.9E-12 1.1E-11 1.7E-11 0. O.

0. 2.5E-11 1.7E-il 9.6E-12-9.4E-12 8.2E-12-2.6E-12-1.6E-11 0. O.

w O. -1.7E-11-1.8E-11-1.4E-11-9.0E-12-4.8E-12-2.5E-12-1.9E-12 0. O.

O. -4.2E-11-3.1E-11-1.8E-11-1.0E-11-5.0E-12-2.4E-12-l'.5E-12 0. O.

O. -1.2E-10-4.2E-11-1.7E-11-7.7E-12-3.4E-12-1.9E-12-1.7E-12 0. O.

! 0. O. O. O. O. O. O. O. C. O.

O. O. O. O. O. O. O. O. O.

' O.

YX COEFFICIENT

0. O. O. O. O. O. O. O. O. O.

O. O. 8.9E-12 1.9E-12-3.9E-14 1.7E-12-5.2E-13-8.0E-12 0. O.

!  ? 0. O. 4.7E-12 4.7E-13 1.0E-12 4.2E-12 4.0E-12-1.6E-12 0. O.

d O. O. -7.5E-12-4.7E-13 4.0E-12 5.5E-12 8.7E-12 1.3E-11 0. O.

1

0. O. -2.7E-11-2.0E-11-1.3E-11-7.2E-12-3.7E-12-2.1E-12 0. O.

O. O. -5.9E-11-2.7E-11-1.3E-11-6.5E-12-3.2E-12-1.9E-12 0. O.

O. O. -4.1E-11-1.5E-11-6.2E-12-2.8E-12-1.3E-12-9.1E-13 0.

'. O.

O. O. O. O. O. O. O. O. O.

' O.

0. O. O. O. O. O. O. O. O. O.

O =

NO. OF PARTICLE MOVES REQUIRE 0 TO COMPLETE THIS TIME STEP 39 l 0 NP =

3131 IMOV =

1

=

TIM (N) 7.88940E+07 TIMV =

2.02292E+06' SUMTCH = 2.02292E+06 i G NP =

3135 IMOV = 2 TIM (N) =

7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 4.04585E+06 0 NP =

3136 IMOV = 3

7.88940E+07 TIM (N) TIMV

2.02292E+06 0 NP = SUMTCH = 6.06877E+06 3139 IMOV = 4

E A TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 8.09169E+06

? 0 NP = 3140 IMOV = 5 2.02292E+06 SUMTCH = 1.01146E+07 d TIM (N) = 7.88940E+07 TIMV =

S 0 NP = 3172 IMOV = 6 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 1.21375E+07

< 0 HP = 3174 IMOV = 7 TIM (N) = 7.88940E+07 TIMV = ' 2.02292E+06 SUMTCH =- 1.41605E+07 0 NP = 3175 IMOV = 8 i

w TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 1.61834E+07 0 NP = 3179 IMOV = 9 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 1.82063E+07

=

0 MP = 3182 IMOV 10 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 2.02292E+07 0 NP = 3184 IMOV = 11 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 2.22522E+07 0 NP = 3200 IMOV = 12 TIM (N) = 7.88940E407 TIMV = 2.02292E+06 SUMTCH = 2.42751E+07 0 NP = 3204 IMOV = 13 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 2.62980E+07 0 NP = 3209 IMOV = 14 7 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 2.83209E+07

$ 0 NP = 3227 IMOV = 15 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 3.03438E+07 0 NP = 3230 IM6V = 16 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 3.23668E407 0 NP = 3230 IMOV = 17 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 3.43897E+07 0 NP = 3246 IM0V = 18 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 3.64126E+07 0 NP = 3250 IMOV = 19 TIM (H) = 7.88940E407 TIMV = 2.02292E+06 SUMTCH = 3.84355E+07 0 HP = 3252 IMOV = 20 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 4.04585E+07 0 NP = 3271 IMOV = 21 TIM (N) - 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 4.24814E+07 0 NP = 3271 IMOV = 22 TIM (h) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 4.45043E+07 0 HP = 3273 IMOV = 23 TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 4.65272E+07 0 MP = 3293 IMOV = 24 a

.E.

7 9 ~ TIK(N) =

7.88940E+07 TIMV = 2.02292E+06 SUMTCH =: 4.85502E+07 i

0 0 NP =

3294 IMOV =

25 i

S TIM (N) = 7.88940E+07 TIMV = 2.02292E+06 SUMTCH = 5.05731E+07 0 NP =

3296 IMOV = 26

< TIM (H) =

7.58940E+07 TIMV- = 2.02292E+06 O NP = SUMTCH = 5.25960E+07 3306 IMOV = 27

=

w TIM (N) 7.88940E+07 TIMV =

2.02292E+06 SUMTCH = 5.46189E+07 0 NP =

3308 IMOV-' = 28

=

TIN (N) 7.88940E+07 TIMV = 2.0?292E+06 0 NP = SUMTCH =. 5.66418E+07 -

3310 IMOV =

29

=

TIM (h) 7.88940E+07 TIMV =

2.02292E+06 0 f4P = SUMTCH =- 5.86648E+07 3322 IMOV = 30

=

TIM (N) 7.88940E+07 TIMV =

2.02292E+06 SUMTCH = 6.06877E+07 0 NP =

3324 IMOV =

31

=

TIM (N) 7.88940E+07 TIMV = 2.02292E+06 0 NP = SUMTCH = 6.27106E+07 3326 IMOV =

32

=

TIM (h) 7.88940E+07 TIMV =

2.02292E+06 0 HP = SUMTCH = 6.47335E+07 1

3350 IMOV = 33

=

n TIM (N) 7.88940E+07 TIMV =

2.02292E+06 i 0 NP = SUMTCH = 6.67565E+07 3354 IMOV =

34

$ TIM (N) = 7.88940E+07 TIMV =

2.02292E+06 0 NP = SUMTCH = 6.87794E+07 3370 IMOV =

35

=

TIH(N) 7.88940E+07 TIMV =. 2.02292E+06 0 HP = SUMTCH = 7.08023E+07 3375 .-1MOV =

36

=

TIM (M) 7.88940E+07 TIMV =

2.02292E+06 0 I4P = SUMTCH = 7.28252E+07 3387 IMOV =

37

=

TIM (N) 7.88940E+07 TIMV =

2.02292E+06 0 NP = SUMTCH = 7.48482E+07 3389 IMOV =

38

=

TIM (N) 7.88940E+07 TIMV =

2.02292E+06 0 NP = SUMTCH = 7.68711E+07 3402 IMOV =

39

7.88940E+07 TIM (N) 'TIMV

2.02292E+06 SUMTCH = 7.88940E+07

'ICONCENTRATIGN i

NUMBER OF TIME STEPS = 1 DELTA T =

7.83940E+07

- , ~ . . _ - . . - - _ . ~ . . ,-. . _ _ . - . . . . - . . . - . . _ - . - , . . _ - . . - .. . . .-

J cvj E

5 . TIME (SECONDS) = ' 7.88940E+07.

? CHEM. TIME (SECONDS) = 7.88940E+07 U CHEM.T1HE(DAYS) . = 9.13125E+02 TIME (YEARS) = 2.50000E+00 S = 2.50000E+00

- CHEM. TIME (YEARS)

< NO. MOVES COMPLETE 0 = 39 0 0 0 0 0 0 0 0 0 0 0 W 0 0 0 0 0 0 0 0 0 0 0 0 0 1- 0 0 0 0 0 0 0 0-0 0 3 1 0 0 0 0 0 0 0 i 0 0 15 4 1 0 0 0 0 0 0 j 0 0 56 15 3 0 0 0 0 0 0 0 0 255 52 8 1 0 0 0 0 0 j

j 0 0 1293 200 25 3 0 0 0 0 0 l 0 0 5356 631 65 7 1 0 0 0 0-

! 0 022925 1541 134 13 1 0 0 0 0 0 0 0 1 1

0 011369 1748 192 20 2 0 0 0 5498 1195 182 24 3 0 0 0 0 0 0 2834 792 149 23 3 0 0 0 10

. 7 0 0 1462 520 116 20 3 0 0 0 0

$ 0 0 856 336 82 16 3 0 0 0

0-0 0

0 0 0 430 193 54 12 2 0 0 0 228 111 35 8  ? 0 O O O 0 0 117 63 22 6 1 0- 0 0 0 0 0 66 37 14 4 1 0 0 0 0 4

O 0 33 21 8 2 1 0 0 0 -0 O O 18 12 5 2 0 0 0 0 0 0 0 11 7 3 1 0 0 0 0 0 0 0 6 4 2 1 0 0 0 0 0

O O 3 2 1 0 0 0 0 0 0

O C 2 1 -1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0 0 0 0 0 0 'O O O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 'O 0 0 0 0 0 0

E m

0 S

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0

." 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CHERICAL MASS BALANCE MASS IN BOUNDARIES = 0.

MASS OUT BOUNDARIES = -4.32657E-02 MASS PUMPED IN = 1.25086E+09 HASS PUMPED OUT = -8.13354E-05 n INFLOW MINUS OUTFLOW = 1.25086E+09 4

w INITIAL MASS STORED = 0.

PRESENT MASS STORED = 3.85120E+07 CHANGE MASS STORED = 3.85120E+07 DECAY OF SOLUTE MASS = -3.39911E+08

= 8.31859E+07 50RBTION STORAGE (S)

SORBTION DECAY (S) = -7.34207E+08 COMPARE RESIDUAL WITH hET FLUX AND MASS ACCUMULATION:

MASS BALANCE RESIDUAL = 5.50488E+07 ERROR (AS PERCENT) = 4.40086E+00 1THALF=600AYS KD=.1 V=1FT/ DAY N=.1 AX=20 AY=4 B=10 SOURCE =1E9/YR 3APR84 0

TIME VERSUS HEA.D AND CONCENTRATION AT SELECTED OBSERVATION POINTS '

PUMP!hG PERIOD NO. 1

{

l

E m

STEADY-STATE SOLUTION 8 0 4

a r

N HEAD (FT) CONC.(MG/L) TIME (YEARS)

< 0 OBS.WELL NO. X Y e

1 2 10 w

0 0.0 0.00 0.000 8.0 ******* .064 1

2 8.0 ******* .128 3 8.0 ******* .192

• • • • • • • .256 I 4 8.0 l

• • • • • • • .321 5 8.0 l 6 8.0 ******* .385 7 8.0 ******* .449 8.0 ******* .513 8

8.0 ******* .577 9' ******* .641 n 10 8.0 4

• • 11 8.0 ******* .705 12 8.0 ******* .769 13 8.0 ******* .833 14 8.0 ******* .897 15 8.0 ******* .962

• • • • • • • 1.026 16 .* 8.0

• • • • • • • 1.090 17 8.'0

• • • • • • • 1.154 18 8.0

• • • • • • • 1.218 19 8.0

• • • • • • • 1.282 20 . 8.0 8.0 ******* 1.346 21

• • • • • • • 1.410 22 8.0 8.0 ******* 1.474 23 24 8.0 ******* 1.538

• • • • • • • 1.603

25 8.0 26 8.0 ******* 1.667

• • • • • • • 1.731 27 8.0 8.0 ******* 1.795 28

a E

=

8 29 E.0 ******* 1.859 4 30 8.0 *******

1.923 ed 31 8.0 1.987

~

0 32 8.0 ******* 2.051 33 8.0 ******* 2.115

& 34 8.0 ******* 2.179 7' 35 8.0 ******* 2.244 36 8.0 ******* 2.308 w

37 8.0 *******- 2.372 38 8.0 ******* 2.436 39- 8.0 ******* 2.500 0 OBS.WElL N0. X 'Y N HEAD (FT) CONC.(MG/L) TIME (YEARS) 2 2 14 0- 0.0 0.00 0.000 1 7.0 0.00 .064 n 2 7.0 0.00 .128 5

3 7.0 36.24 .192 4 7.0 216.37 .256 5 7.0 331.65 .321 6 7. 0. 548.47- .385

'7 7.0 928.14 .449 8 7.0 982.51 .513 9 7.0 1466.19 .577 10 7.0 1485.24 .641 11 7.0 1375.83 .705 12 7.0 1670.89 .769 13 7.0 1553.58 .833 14 7.0 1476.12 .897 l 15 7.0 .1349.46 .962 16 7.0- 1533.03 1.026 17 7.0 1538.69: 1.090 18 7.0 1631.56 1.154 19 7.0 1672.57 1.218 20 7.0 1727.07 1.282 21 7.0 1581.62 1.346

'I 4

E

%* 22 - 7.0 1707.25 1.410

• 23 7.0 1577.20 1.474 5 24 7.0 1471.99 1.538-

" 25 7.0 1512.43 1.603 26 7.0 1626.47 1.667 e 7.0 1609.57 1.731 S 27

- 28 7.0 1661.53 1.795 w 29 7.0 1709.65 1.859 30 7.0 1616.11 1.923 31 7.0 1582.09 1.987 32 7.0 1555.08 '2.051 33 7.0 1401.51 2.115 34 7.0 1521.43 2.179 35 7.0 1407.56 2.244 36 7.0 1500.16 2.308 37 7.0 1531.35 2.372 38 7.0 1567.67 2.436 39 7.0 1462.03 2.500

'7 0 OBS.WELL N0. X Y N HEAD (FT) CONC.(MG/L) TIME (YEARS) l @ 2 20 3

0 0.0 0.00 0.000 1 5.5 0.00 .064

! 2 5.5 0.00 .128 3 5.5 0.00 .192 4 5.5 0.00 .256 5 5:5 0.09 .321 6 5.5 0.00 .385 7 5.5 .01 .449 8 5.5 .12 .513 9 5.5 .60 .577 10 5.5 1.60 .641 11 5.5 3.59 .705 12 5.5 7.13 .769 13 5.5 11.35 .833 14 5.5 11.99 .897

~

E h 15 5.b 15.13 .962 4 16 5.5' 22.50 1.026

- 17 5.5 25.15 1.090 S 18 5.5 24.39 1.154 19 5.5 30.70 1.218 g 20 5.5 30.83 1.282

! r 21 5.5 34.03 1.346

" 22 5.5 34.74 1.410 23 5.5 28.94 1.474 24 5.5 31.59 1.538 25 5.5 32.79 1.603 26 5.5 33.86 1.667 27 5.5 35.51 1.731

, 28 5.5 29.74 1.795 f 29 5.5 37.63 1.859 30 5.5 34.09 1.923 l 31 5.5 33.74 1.987 i 32 5.5 33.38 2.051 33 5.5 33.39 2.115 n 34 -5.5 35.67 2.179 s'd 35 5.5 32.18 2.244 36 5.5 35.28 2.308 37 5.5 33.08 2.372 38 5.5- 33.43 2.436 39 5.5 33.02 2.500 0 GBS.WELL NO. X Y N HEAD (FT) CONC.(MG/L) TIME (YEARS) 4 2 30 0 0.0 0.00 0.000 1 3.0 0.00 .064 2 3.0 0.00 .128 3 3.0 0.00 .192 4 3.0 0.00' .256 5 3.0 0.00 .321 6 3.0 0.00 .385 7 3.0 0.00 .449

E A

?

D 8 3.0 0.00 .513 S 9 3.0 0.00 .577

!' < 10 3.0 0.00 .641 S 11 3.0 0.00 .705 12 3.0- .00 .769 (d 13 3.0' .00 .833 14 3.0 .00- .897.

15 3.0 .00 .962 l 16 3.0 .00 1.026 17 3.0 .00 1.090 18 3.0 .00 1.154 19 3.0 .00 1.218 20 3.0 .00 1.282 21 3.0 .00 1.346 22 3.0 .00 1.410 ,

23 3.0 .00 1.474 24 3.0 .01 1.538 n 25 3.0 .01 1.603 i

$ 26 3.0 .02 1.667 27 3.0 .02 1.731 28 3.0 .03 1.795 29 3.0 .04 1.859 30 3.0 .04 1.923 31 3.0 .05 1.987

' 32 3.0 .04 2.051 33 3.0 .05 2.115 34 3.0 .05 2.179-l

.05 2.244 35 3.0 36 3.0 .06 2.308

' 37 3.0 .06 2.372 38 3.0 .06 2.436

' 39 3.0 .07 2.500 i

APPENDIX D ANALYTICAL SOLUTIONS FOR RADIONUCLIDE TRANSPORT IN GROUNDWATER Analytical solutions are available for some forms of radionuclide transport e'quations. Typically, these solutions are for steady-state flow systems with either uniform (spatially constant) velocities or simply varying non uniform velocities. Aquifer properties are considered uniform and the aquifer system is treated as an infinite or semi-infinite medium. Although real aquifers do not have these ideal characteristics, analytical solutions (.an approximate the behavior of more complex systems and can be used to perform preliminary and order-of-magnitude analyses. Moreover, in the absence of site-specific data (such as aquifer geometry and spatial variability), analytical solutions are simple and cost effective.

Several analytical solutions are presented below for one-dimensional and two-dimensional radionuclide transport in groundwater (Table D-1). This collection is not comprehensive, but it does include examples of the range of conceptual models for which analytical solutions are available. Grove and Kipp (1981) briefly review the contaminant transport modeling literature and reference solu-tions by many investigators. Bear (1972, 1979) presents solutions for a variety of boundary conditions and reaction terms in one and two dimensions and in radial and transformed coordinate systems. Baetsld (1969) and Codell and others (1982) illustrate application of analytical solutions for radionuclide migration in groundwater.

ONE-DIMENSIONAL FLOW AND TRANSPORT l

A one-dimensional conceptualization of radionuclide transport ignores the spread-i I

ing of the concentration plume perpendicular to the flow streamline. This con-ceptual model can include reaction terms and dispersion in the direction of flow as well as a variety of boundary conditions. One-dimensional analyses are used most often in the determination of system transport properties from experimental data in laboratory columns. Given a lack of field data, especially transver dispersion data, a one-dimensional analysis estimates radionuclide migration neglecting the reduction in peak concentrations from lateral spreading. In this sense, one-dimensional analysis is generally considered more conservative than two-dimensional analysis.

Bear (1972, 1979), Gershon and Nir (1969), and others present solutions for one-dimensional advection-dispersion with various reaction terms and boundary condi-tions. In particular, Gershon and Nir (1969) discuss the sensitivity of con-centration profiles to different boundary conditions. The governing equation for transport of a retarded (by sorption) radioactive solute in one-dimensionai l incompressible flow is (Bear, 1972) l l

0 a R D 30 -V 3ax0 - R AC d at. ax ax d (D-1)

NUREG-1101, Vol. 3 D-1

Table D-1 Overview of representative analytical solutions for radionuclide transport in groundwater Conceptual model Special considerations ONE-DIMENSIONAL No lateral sp, reading Uniform flow No recharge / discharge along flowpath

1. Leach source Exponential source, no dispersion

2. Instantaneous source Longitudinal dispersion

3. Fixed concentration source Similar to 2, error function solution Non-uniform flow Recharge along flowpath

4. Leach source .

No dispersion, similar to 1

5. Transport without mixing Separate contaminated layer TWO-DIMENSIONAL Horizontal, no vertical processes Uniform flow No recharge / discharge along flowpath Similar to 2 4

~6. Instantaneous point ~ source i 7. Instantaneous line source Line normal to flow l 8. Instantaneous area source Rectangular 4

9. Constant injection point source Convolution of 6 i

in which Rd [-] = retardation C[M/L3] = solute concentration D[L2/T] = the longitudinal dispersion coefficient V[L/T] = flow velocity, which may vary in time and space A[T 1] = the radioactive decay rate ,

The retardation factor R approximates the effect of reactions which remove

, d j solute from the flowing fluid. For linear equilibrium sorption on the solid matrix, the retardation factor is (Freeze and Cherry, 1979)

(1 - n)p R =1+

5 K (~}

d n d j

where n[L3/L3] = porosity p [M/L3] = solid matrix density (solid mass per unit solid volume) s

Kd [L /M] = the distribution coefficient NUREG-1101, Vol. 3 0-2

For typical groundwater flow situations, molecular diffusion is insignificant and the dispersion coefficient can be written (Freeze and Cherry,1979)

D=aVx (D-3) in which ax [L], the longitudinal dispersivity, is a characteristic length of the dispersion process.

Since the governing equation D-1 is linear in C, solutions for arbitrary source strengths can be determined from instantaneous impulse response solutions using the convolution integral as described below (equation D-22, for 2-D horizontal flow).

Uniform Flow In a homogeneous aquifer with relatively small local recharge and discharge in the area of interest, the flow field may often be approximated as uniform con-stant velocity. Velocity V and dispersion D in equation D-1 are constant for this case.

(1) Leach source - no dispersion Neglecting longitudinal dispersion, the equation for one-dimensional radioactive solute transport becomes Rd Eat= -V Eax- R ACd (D-4)

Initially, C = 0 at all locations (x). An exponentially decaying leaching source boundary condition is m'A C (x = 0, t > 0) =

nbV **P -(A

• AL )t (D-5) where i

AL [T 1] = the source leach rate decay coefficient m'[M/L] = the source mass per unit width Units of solute mass and radioactivity, i.e., curies, are considered inter-changeable in this appendix. The concentration at the source is dropping because of both a decreasing leach rate and radioactive decay of the source inventory.

Using the method of characteristics, the solution to equation D-4 is C(x, t) = 0 when t < xRd m'A xR AxR xR L d d d C(x, t) =

nbV eXP -(A + AL ) t - V V_

when t > y (D-6)

NUREG-1101, Vol. 3 D-3

Since the source-strength is dropping for all t > 0, and dispersion is ignored, the peak concentration C occurs when the advected front first reaches any loca-tion (x). Substituting t = xR d/V into equation D-6 gives the peak concentration for any location

. m'A ~ AxR ~

• • P ^

(D-7)

C(x) = nbV _ V _

(2) Instantaneous source A slug of radioactive solute mass per unit width m'[M/L] is instantaneously injected at the point source location, x = 0. Initially, solute concentration is zero for all locations (x). Governing equation D-1 includes dispersion and the boundary condition is C(x = 1 m, t) = 0 (D-8)

The corresponding analytical solution is (Bear 1979; Crank, 1956) m'

~ (x-Vt/Rd)

) **P - At (D-9) nb(4nax Vt/Rd ) -

_ 4a xVt/R d in which b[L] is aquifer thickness. Figure D-1 is a graphical representation of the concentration profile in relation to the Plume center which moves at retarded velocity V/Rd. The peak concentration C as a function of distance from the source is approximated by substituting x = Vt/Rd and t = xRd /V into equation D-9 to obtain:

^

• (D-10)

C(x) = exp[-AxRd nb(4nax *)b The time of peak, t = xR d/V, and the drop in concentration because of decay both increase with increasing retardation. The exact peak occurs earlier and this approximation is not suitable for radionuclides with rapid decay rates or short half-lives.

(3) Fixed concentration source A widely applied one-dimensional advection-dispersion solution is that for a semi-infinite system wherein the concentration at x = 0 is fixed at a constant value. The boundary conditions applied to equation 0-1 for this case are C=0 at x = =

C=C at x = 0 (D-11) g i

NUREG-1101, Vol. 3 0-4

1.0

/\

m l \"

0.5 o #

t,>t 2

0 3 w -

5 -4 -3 -2 -1 0 1 2 3 4 5 X - Vt/ R d Figure D-1 Relative concentration for a slug of non-decaying solute (after Bear, 1979).

l l

NUREG-1101, Vol. 3 0-5

and the corresponding solution is (Bear, 1979)

( x-2paxVt/R d C(x,t)=[C explx 3 (exp{-xp} erfc (4"xVt/Rs )b _

x+2pax Vt/R d

+ exp{xp} erfc (4a Vt/Rd)b .

where p2 = \*+AR d "x Y

\ x/

and erfc is the complimentary error function

=

2 erfc(y) = 1 exp( X2 )dx (D-13) erf(y) = /d y where erf is the error function. The error function and complimentary error function are tabulated by Abranowitz and Stegun (1964) and Freeze and Cherry (1979).

Without radioactive decay, A = 0, and the solution (equation 0-12) becomes (0gata and Banks, 1961)

F x - Vt/Rd ~

+ expl x\lerfc x + Vt/R d" C /

o C(x, t) = y erfc (D-14)

(4ax Vt/Rd)y. \* - (4"xVt/Rd) .

Figure D-2 is a graphical representation of equation D-14. For the fixed con-centration boundary, concentrations at all locations (x) increase monotonically to a steady-statre value. In the absence of decay, the steady-state solution is C = C , for all locations (x).

g Non-uniform fiow Recharge to the groundwater flow system usually occurs along the flowpath.

This recharge has several impacts, including increasing dilution, and increasing velocities, thus reducing travel time.

(4) Leach source - no dispersion Following Codell (personal communication, 1984) a phreatic (unconfined) ground-water flow system is conceptualized as shown in Figure D-3. All flow is from 1ccal uniform recharge and both the discharge and saturated thickness vary in space. The radioactive solute transport equation without dispersion (equa- ,

tion D-4) is applied with a source boundary condition identical to that above for an exponentially decaying source in uniform flow (cf., equation D-5). The analytical solution to equation D-4 for concentration is (Codell, personnel communication, 1984; cf., equation D-6 supra)

NUREG-1101, Vol. 3 D-6

0.999 a;/X oo gI Ib \0Y, a

20 50 /

0.8 a x/X: #

i 100 [

[/

\ ! / 50 20 10 0.1 1 J '

l '

!/ 0.5 f 02 0.1 0.00, /

0.05

/

0.1

//n) 0.5 1.0 5.0 10 50 Vt/Rdx i

Figure D-2 Relative concentration for one-dimensional advection-dispersion I

with adsorption (after Ogata and Banks, 1961) l l

l l

l l

NUREG-1101, Vol. 3 D-7

Groundwater Source Stream-Discharge Divide N Location N Well Location f i r l, r , r , , , r f i r , r l,r i r , r , r I l

..-,a.e,<,,... l I

s ff?g># 2 _ . . . hk u l l p -= E F ~ mi,M}T?lg(d?;:fni {?lj.g x ~~

I

=d%hs  !

i -- i

{

y=:;=

m =:

i HM5

t" -~ ,

ll l Y YllYllYll$NWlWllWll&

I I I I X=X d X=0 X=X g Figure D-3 Schematic diagram for a non-uniform flow system I

1 NUREG-1101, Vol. 3 D-8

C(x, t) = 0 when t < t R (0-15)

C(x, t) = m'A' exp - (A + A )(t - t ) - At when t > t L R R R Q

where the travel time to location x, tR [T], is given by

~

t =

ns fx xg /-xd t( 3 1 ( g /_ (0-16a) in which the length factor A[L] is A= 'h (xH ~ *d) *RH2 _

(0-16b) where H[L] is the saturated aquifer thickness at the stream and the function 1(z) is defined T(z) = (1 z2 )h - fn 1 + (1 z }

(0-16c)

Q[L2/T] = the aquifer discharge per unit width at location x K[L/T] = the hydraulic conductivity N[L/T] = the recharge rate The aquifer discharge Q is the recharge rate N times the distance from the groundwater divide to x.

As above, the peak concentration is given by equa-tion D-15 with t c t ,R when the advected front first reaches x.

(5) Transport without mixing Ostendorf and others (1984) consider uniform recharge which does not mix witt' the contaminated groundwater but flows on top of the contaminated zone as an immiscible layer. Local recharge affects the flow velocity but does not dilute the concentration in the contaminated zone. As above, this conceptual model ignores dispersion.

At the source, the velocity V s[L/T] is simply the groundwater discharge per unit width q[L2/T], which includes upgradient sources, divided by porosity n [L3/L3] and saturated thickness h3 [L]. Ostendorf and others (1984) present a near-field technique for predicting V and s the source concentration C EH/b 3" s

The time of arrival t, required for solute to reach an observation point is (Ostendorf et' al. ,1984) t a"V s

( }

NUREG-1101, Vol. 3 D-9

where Rd [-] = retardation x = the distance from the source, and Nh q (0-18)

Y=qs8 + d -s tan p in which p[L/L] is the slope (small) of the aquifer bottom The concentration in the contaminated groundwater layer is (Ostendorf et al. ,1984)

C=C seXp -

t (0-19) a, d

The concentration given by equation D-19 is higher than that given by equation D-15 because of the dilution from recharge considered in developing equation D-15. In using equation D-19 to estimate concentrction in a well, it is assumed that the well draws water only from the contaminated layer, not from the entire saturated thickness.

TWO-DIMENSIONAL HORIZONTAL FLOW AND TRANSPORT Many groundwater flow systems can be approximated as a uniform horizontal flow field of. infinite extent. Vertical flow and solute transport are assumed to be insignificant in determining the concentration in a water well downgradient.

Groundwater velecity is constant (no local recharge) and is in the x direction.

This assumption also ignores the effect of the source flux on the flow field configuration.

The governing equation for the two-dimensional advection-dispersion of a radioactive solute in uniform groundwater flow can be written (Bear, 1979) a _y a (D-20)

R

,yy - R AC d

da * "xV y in which C[M/L3] = concentration Rd [-] = retardation V[L/T] = groundwater velocity in the x direction a [L] = longitudinal dispersivity x

a [L] = transverse dispersivity A[T 1] = radioactive decay rate The lefthand-side term is the change in storage term where Rd incorporates the change in storage of solute adsorbed on the solid matrix. The first two terms on the righthand side (RHS) represent dispersion along the flowpath (x) and normal or transverse to the flowpath (y), respectively. The third RHSThe term is final change in advection, or solute carried with the flowing groundwater.

NUREG-1101, Vol. 3 D-10

RHS term is radioactive decay, including decay of the adsorbed solute. This formulation assumes that sorption can be represented by a constant retardation such as p (1 - n)

S R

d

=1+ n d (D-21) in which ps [M/L ] = the solid matrix density (solid mass per unit solid volume) n[-] = porosity Kd [L /M] = the linear distribution coefficient In addition, it is assumed that the dispersion coefficient D[L2/T] is propor-tional to velocity Dxx " "xV O

yy =aV y The governing equation is linear and the solution, for an arbitrary source strength as a function of time, is the integral over time of the source strength times the unit impulse response. The unit impulse response is the solution for an instantaneous release of a source strength (mass) of unity. The integral over time for an arbitrary source is the convolution integral (Dooge, 1973) t C(x, y, t) = f(t)C g (x, y, t t)dt (D-22) where

( f = the source strength as a function of time C9 = the unit impulse response The unit impulse response is independent of source strength and depends on aqui-fer properties and the spatial characteristics of the source. Three cases are considered below: point source, line source parallel to y, and rectangular area source (cf., Codell et al., 1982; Bear, 1972; Carslaw and Jaeger, 1959).

(6) Instantaneous point source For a unit slug solute mass injected instantaneously at a point x = y = 0. the unit impulse response C gp [M/L3] is (Codell et al., 1982) y " (x-Vt/Rd) Y 2

Cg p (x, y, t) = exp _ -

- At (D-23) 4nnbtV(a a )g h xVt/R d xy _

y Vt/R d -

where '

n[-] = porosity b[L] = aquifer thickness NUREG-1101, Vol. 3 0 - l'.

Both n and b are constant in space. At any given distance (x) downgradient from the source, highest concentrations occur at the plume centerline y = 0.

The centerline unit impulse response is 1 ~ (x-Vt/Rd )

Cj p (x, y = 0, t) = exp _

- At (0-24) 4nnbtV(a x "y)y -

4"xVt/R d -

(7) Instantaneous line source For a line source parallel to the y axis centered at x = y = 0 and extending from y = -w/2 to y = w/2, the unit impulse response C;f[M/L3] is (Codell et al., 1982) 1 - (x-Vt/Rd)

Cgg(x, y, t) = exp _

- At R )g 4nbwR d (""x d 4"xVt/R d -

(0-25) erf

• lI

• Y + erf

• l ~Y (4ay Vt/Rd ) (4"yVt/Rd ) -

where erf is the error function (Abramowitz and Stegun, 1964; Freeze and Cherry, 1979). For the centerline unit impulse response, the last bracketed term becomes w2 (0-26) 2 erf

_ (4ay Vt/Rd ) -

As source width decreases, w 0, the line source unit impulse response converges to that for a point: C;g4C j. p (8) Instantaneous area source For an area source centered at x = y = 0 and extending from x = -2/2 to x = f/2 and y = -w/2 to y = w/2, the unit impulse response Cia [M/L ] is (Codell et al.,

1982) l x + f/2 - Vt/R d x - 1/2 - Vt/R d 1

erf - erf Cia (x, y, t) = 4nbwgR d _ (4a xVt/Rd) -

- (4"xVt/Rd) -

(D-27) w2+y w2-y erf + erf exp[-At]

, _(4a y Vt/R d )- -( 4"yVt/Rd )

The centerline solution is obtained exactly as above in the line source solution (equation D-26). As source length decreases, 140, the area source unit impulse As f+0 and w 0, the response converges to that for a line source, Cia4C j. p 4 solution converges to the point source solution, Cia4C$p.

NUREG-1101, Vol. 3 0-12

The peak centerline concentration is highest for a point source and lowest. for an area source. At large distances from the source, the differences between the three solutions become negligible. This distance depends on the disper-sivities and the source dimensions. For example with a transverse dispersivity of 25 ft and a source width of 150 ft, the peak concentration at x = 1000 ft for a line source solution is about 98% of the corresponding peak concentration for a point source solution (9) Constant injection point source For an arbitrary source function, the convolution integral (equation D-22) must be evaluated numerically (cf., Codell et al., 1982). For certain special source functions, however, equation D-22 can be evaluated analytically.

For a continuous constant injection source function (f(t) = f') at a point (x = y = 0), the analytical solution to equation D-22 is (Wilson and Miller, 1978, 1979; Fried, 1975)

C(x, y, t) = f' exp(xB) W(u,r/B) (D-28) 4nnbV(ox "y) in which B = 2a x

r2

"

• 4ya Vt/R x d r= x2+ y2 y b Y = 1~ + (2BAR d )

W(u, r/B) = fexp -

X+

dX 4B"X where W(u, r/B) is Hantush's leaky well function (Hantush,1964). The leaky well function has been tabulated by several authors (Hantush, 1964; Walton, 1970; Miller, 1975). Wilson and Miller (1978) discuss a graphical solution technique using equation D-28 and approximate solutions for large distances from the source.

For large r/B > 1, where r is the weighted radial distance, the steady-state (t+=) concentration is approximately (Wilson and Miller,1978) f(x, y) = f' exp(x/B) exp(-r/8) (D-29)

(8n/B)bnbV(o x "y) (")

As above, at any distance (x) downgradient from the source, the maximum concen-tration occurs at the plume centerline (y = 0). In the absence of decay, A = 0 NUREG-1101, Vol. 3 0-13

cnd y = 1. For this case the weighted radial distance is r' = (x2 + y 2ax/"y) cnd on the centerline r' (y = 0) = x Thus, from equation D-29 the steady-state concentration for a nondecaying solute on the plume centerline at large r/B > 1 is C'(x, y = 0) = A(x)5 (D-30) where A=

(8n/B)b nbV(axya )5 has no effect on the steady-state It should be noted that retardation Rd concentration distribution in the absence of decay.

SUPMARY Several analytical solutions for radionuclide transport in groundwater have been described. Table D-1 summarizes the various solutions and the conceptual models which they represent. These solutions can be used to estimate potential groundwater contamination from radioactive waste burial. Criteria for applying these solutions include: the appropriateness of the conceptual model, the available data and solution data requirements, and the degree of uncertainty or conservatism of solution input and results.

REFERENCES Abramowitz, M. A.. and I. A. Stegun Handocok of Mathematical Functions. U.S.

Dept. of Commerce, National Bureau of Standards, AMS55, 1964. .

Baetsid, L. H.. " Migration of Radionuclides in Porous Media'" pp. 707-730 in A. M. F. Duhamel (ed.), Progress in Nuclear Engineering, Series XII, Health Physics, Pergamon Press, Elmsford, New York, 1969.

Bear, J., Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.

Bear, J., Hydraulics of Groundwater, McGraw-Hill, New York, 1979.

Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, Oxford University, Press, London, 1959.

Codell, R. B. , K. T. Key, and G. Whelan, "A Collection of Mathematical Models for Dispersion in Surface Water and Groundwater, U.S. Nuclear Regulatory Commission, NUREG-0868, 1982.

NUREG-1101, Vol. 3 0-14

I i

l Crank, J., Mathematics of Diffusion, Oxford University Press, New York and l London, 1956. '

i Dooge, J. C. I., " Linear Theory of Hydrologic Systems," Technical Bulletin 1468, U.S. Dept. of Agriculture, Washington, DC, 1973.

Freeze, R. A., and J. A. Cherry, Groundwater, Prentice-Hall, Englewood Cliffs, New Jersey, 1979.

Fried, J. J., Groundwater Pollution, Elsevier, Amsterdam, The Netherlands, 1975.

Gershon, N. D. , and A. Nir, " Effects of Boundary Conditions of Models on Tracer Distribution in Flow Through Porous Mediums," Water Resources Research, 5(4):830-839, 1969.

Grove, D. B., and K. L. Kipp, "Modeling Contaminant Transport in Porous Media in Relation to Nuclear-waste Disposal: A Review," pp. 43-63 in C. A. Little and L. E. Stratton (eds.), Modeling and Low-Level Waste Management: An Interagency Workshop (NTIS CONF-801217), Oak Ridge National Laboratory,1981-Hantush, M. S., " Hydraulics of Wells," pp. 281-442, in V. T. Chow (ed.),

Advances in Hydroscience, 1, Academic Press, New York, 1964.

Miller, P. J. , "Modelling Local Groundwater Contamination," unpublished M.S.

thesis, Massachusetts Institute of Technology Dept. of Civil Engineering, 1975.

Ogata, A., and R. B. Banks, "A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media., U.S. Geological Survey Professional Paper 411-A, 1961.

l Ostendorf, D. W. , R. R. Noss, and D. O. Lederer, ' Landfill Leachate Migration Through Shallow Unconfined Aquifers," Water Resources Research, 20(2):291-296, 1984.

i Walton, W. C., Groundwater Resource Evaluation, McGraw-Hill, New York, 1970.

l Wilson, J. L., and P. J. Miller, Two-dimensional Plume in Uniform Groundwater Flow," J. Hydraulics Division, ASCE, 104(HY4):503-514, 1978.

Wilson, J. L, and P. J. Miller, "Two-dimensional Plume in Uniform Groundwater Flow, Closure," J. Hydraulics Division, ASCE,105(HY12):1567-1570,1979.

NUREG-1101, Vol. 3 D-15

4 i

j APPENDIX E j~ SENSITIVITY OF RADIONUCLIDE TRANSPORT TO GROUNDWATER VELOCITY Estimated peak concentrations in groundwater downgradient from a radionuclide source, for example on a plume centerline or at the center of- mass' of a spread-

! ing slug, are affected by the velocity used in calculations. Velocity deter-

, mines the travel time of radionuclides from the source to a receptor during which radioactive decay occurs. In addition, velocity is often considered di-l -rectly proportional to flux, the amount of flow through the aquifer. The in-

?

jected source mass is diluted by this through-flow, thus higher velocity indi-cates nigher flux and reduced concentration at the source, as well as at the receptor downgradient. These two effects, decreased decay and increased dilu-tion with increasing velocity, counteract each other and a critical velocity can be determined, for certain conceptual models, which will result in maximum 1 l

4 peak concentration for a given set of system parameters. '

A critical-velocity exists only if flux is considered proportional to velocity and if the radionuclide decays. In many case from recharge or pumping measurements and be can,s, groundwater fixed as a constant. flux can be estimated Since the dilution is then fixed, higher velocities result in less decay and higher concentrations, and no' critical velocity exists. However, in this case, the velocity can also be estimated from flux and porosity estimates.

ONE-DIMENSIONAL CONCEPTUAL MODEL A simple conceptualization of radionuclide transport in groundwater is one-dimensional advection in a uniform flow field with linear equilibrium adsorption

! (retardation) and radioactive decay. This conceptual model_ ignores dispersion.

l Initially, concentration is zero for all locations (x). At time t = 0, radio-nuclide mass (measured as activity) is injected at a constant rate m[Ci/T/L]

per unit width into the system at x = 0. Ahead of the advected front, concentra-tion remains zero, since dispersion is ignored. At and behind the advected front, the concentration is:

/ xAR C(x) = n V **P f V frx1 (E-1) where C[Ci/L3] = radionuclide activity concentration V[L/T] = uniform groundwater velocity in the x direction b[L] = aquifer thickness '

n[L3/L3] = porosity Rd [-] = retardation coefficient A[T 1] = radioactive decay rate

-NUREG-1101, Vol. 3 E-1

_ _ _ _ . _ _ . . ~ . _ _ . _ . _ _ _ _ _ . _ _ _ _ _ _ _ _ . _ ___ _ _ . _ _

l 5 t

4 The sensitivity of concentration'to velocity is the derivative of equation E-1 with respect to V f r xAR aC _ jxAR d 13 dT (E-2)

, g- ys 92 exp - y fxAR d i i = -

C At a local maximum, this derivative equals zero. For a non-trivial solution t C / 0, thus setting equation E-2 equal to zero, yields

.V = xAR (E-3) c d where V = the critical velocity for maximum concentration c

This critical velocity depends on the distance from the source at which concen-

- trations are estimated (i.e. , the receptor location), the decay rate, and the retardation coefficients. The parameters for an example problem are shown in

Table E-1.. Table E-2 illustrates the variation of concentration for several velocities at-two locations. For this case, an order-of-magnitude underestimate of velocity results in much lower concentrations than an order-of-magnitude 3

overestimate.

Table E-1 Parameters for example problems i Parameter Symbol Value Porosity n 0.1 Thickness b 10 ft Source strength m 1 Ci/yr "etardation R 10 d 0.021 yr 1 Decay rate A Malf-life 33 yr I TWO-DIMENSIONAL CONCEPTUAL MODEL f A common conceptual model includes dispersion, or spreading of the injected radionuclide, both along the flowpath (longitudinal) and perpendicular to the l flowpath (transverse). When dispersion is considered, the critical velocity is j not equal to that in equation E-3 because longitudinal dispersion essentially decreases the effective travel time, moving some radionuclides faster than the i groundwater.

i NUREG-1101, Vol. 3- E-2 i

b.

Table E-2 Calculated concentrations

• V(ft/yr) C(x = 100 ft) C(x = 1000 ft)

Ci/ft8 1 7.58E-10 6.28E-92 2.1 2.16E-6 1.77E-44 10 0.0122 7.58E-11 21 0.0175 2.16E-6 l 100 0.0081 1.22E-3 210 0.0043 1.75E-3 1000 9.79E-4 8.11E-4  ;

, 2100 4.71E-4 4.31E-4

• Ve (x = 100) = AR d x = 21 ft/yr Vc (x = 1000) = 210 ft/yr Wilson and Miller (1978, 1979) developed an approximate solution for the steady-state concentration plume from a constant point source at rate m' [Ci/T]. This solution considers both longitudinal and transverse dispersion. The governing equation for this conceptual model is a .yy .y a R

d * "xV y - R AC (E-4) d x y x in which ax and ay [L] are the longitudinal and transverse dispersivities, respectiva1y. For r/8 > 1, the approximate solution is (Wilson and Miller,

, 1978):

i C(x,y) = (E-5) nbV 8nax "y (r/B)S where f a x

r= i x2+ a y 2) y 4

\ Y } '

E = la, 2BAR d

y=1+ y NUREG-1101, Vol. 3 E-3

The sensitivity of the centerline (y = 0) concentration to velocity is xAR ad = _1}_xARd. d C (E-6)

S S S I

Vy 2Vy(xyB)_.)

Setting equation E-6 equal to zero, the critical velocity V' is xAR d 8 (E-7)

V'=(1+2BAR/V')y, d -

2(1+BAR/V')S, x d This equation must be solved iteratively, since V' is on both sides of the equation.

Substituting Vc = xARd , the one-dimensional advegtior, critical velocity from equation E-3, equation E-7 can be written

= (E-8) 1+B/2x(1+2BV/xV')-b(1+2BV/xV')~b c c The ratio of the critical velocity for the two-dimensional advection-dispersion model to the critical velocity for the one-dimensional advection model is a function of x/B only (Figure E-1). This term (x/8 = x/2ax) is the dimension-less distance from the source.

An approximation to the critical velocity for the two-dimensional model can be developed by examining the relationship of the one-dimensional critical velocity (equation E-3) to the analytical solution (equation E-1). Substituting equa-tion E-3 into equation E-1, the power of the exponent is -1. The approximate analytical solution for the two-dimensional model (equation E-5) has a form similar to equation E-1, including V in the denominator. Setting the power of the exponent in equation E-5 to -1 results in 2BAR y _ d (E-9) I a ~ 2(B/x) + (B/x)z where V, is an approximate critical velocity for the two-dimensional case. For large x/8, which is implied by large r/B, the second term in the denominator in equation E-8 is small, and V, reduces to the one-dimensional critical velocity Vc (equation 3) in the limit.

Figure E-1 tion E-7) toshows the ratio of the critical the one-dimensional two-dimensional velocity Vccritical velocity)V' (equation E-3 and (equa-to the approximate two-dimensional critical velocity V, (equation E-8) as a function of distance from the source. Va is a better approximation of V' than Vc '

NUREG-1101, Vol. 3 E-4

1.0 ---

.r#,.-

/

t 0.8 -

l

/

V'/Vap#

O l' V'/V c

{cc 0.6

,/

/

f a __,s 30.4 -

W 0.2 -

' '"I ' "I ' "I ' ''I ' ' ' ' ' ' '

0.0 O.01 0.1 1.0 10 100 1000 x/B - DIMENSIONLESS DISTANCE Figure E-1 Ratio of critical 2-D velocity (V') to an approximate critical velocity (Va) and to the critical 1-D velocity (V )c NUREG-1101, Vol. 3 E-5

because it incorporates some of the effect of dispersion. Recalling that B = 2ax , V,is about 90% of V' for x = 2a x, and V,is greater than 99% of V' for x = 20ax*

The variation of steady-state centerline concentration with velocity is shown in Figure E-2. The exact analytical solution (to which equation E-5 is an approximation) (Wilson and Miller 1978,1979) is used for a case governed by the parameters in Table E-1. In addition, the longitudinal and transverse dis- ,

persivities are 20 ft and 4 ft, respectively. The effect of underestimating i velocity is qualitatively greater than an overestimate. As distance from the source increases, the differences in concentrations calculated with the different critical velocities becomes negligible.

SUMMARY

For certain screening models of radionuclide transport in groundwater, a criti-cal velocity can be determined which will result in maximum calculated concen-trations. This critical velocity is determined by the distance from the source, the radioactive decay rate, the retardation coefficient, and the longitudinal dispersivity. Critical velocities can be determined for other conceptual models and corresponding analytical solutions. For many cases, equation E-3, the criti-

, cal velocity for a simple one-dimensional advection model, may provide a reason-able estimate which can be verified by computing solutions for higher and lower velocities.

REFERENCES Wilson, J. L., and P. J. Miller, "Two-dimensional Plume in Uniform Groundwater Flow," J. Hydraulics Divison, ASCE, 104(HY4):503-514, 1978.

Wilson, J. L. , and P. J. Miller, "Two-dimensional Plume in Uniform Groundwater Flow, Closure," J. Hydraulics Divison, ASCE, 105(HY12): 1567-1570, 1979.

l 5

NUREG-1101, Vol. 3 E-6

10~1 10~8 -

x = 10 ft Z 10-5 _

g x =100 ft '

m W -

g 107 x =1000 ft u

z O

O -

10~8 10~11 I I ' '

O.01 0.1 1.0 10 100 1000 VELOCITY (ft /yr) t Figure E-2 Steady-state centerline concentration vs. velocity NUREG-1101, Vol. 3 E-7

groau s= o s. =uctuan 1uLaroav co== asio= i tiro,.1~u-sa m c rioc. u ,v.. , ,,

E*i 3" ' E BIBLIOGRAPHIC DATA SHEET see msrauctio=s o rai aaviais NUREG-1101, Vo1. 3 2 rirLE A=o suerirLE 3 LE AVE ELANK onsite Disposal of Radioactive Waste Volume 3: Estimating Potential Groundwater a pare aeroar co=*Leveo Contamination wo=ra j vaaa e curaoais. Augus[ 1986 Daniel J. Goode. Stanley M. Neuder, f . oare ascoar issueo Roger A. Pennifill, Timothy Ginn g*oara ve^a l

Ngvember 1986 i ,a,. cauma onaa :avio= 4=a .=o =4 u=o mooness r, ,,,,i, c,.,

e e7,acrirasnewoax unir uuusea Division of Waste Manage ent I' ' 54112 Office of Nuclear Materi Safety and Safeguards ["'"'"'*""**"

U.S. Nuclear Regulatory C ission Washington. DC 20555 [

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U.S. Nuclear Regulatory Commis ion Washington, DC 20555 in sue,Lawaraa, notes 13 A se a AC1 (200 war,s e, ress, 7 Volumes 1 rnd 2 of this report describe he h 's methodology for a wessing the potential public hellth and environmental impacts soc ted with onsite disposal of very low activity radioactive materials. This vol e (Vol. 3) describes a general methodology for predicting potential groundwater contamina n from onsite disposal. The methodology includes formulating a conceptual model, r esenting the conceptual model mathematically, cetimating conservative parameters, and pr 'd ting receptor location. An example case study of shallow burial of waste containi - i ne-125 illustrates application cf the M0CMOD84 version of the USGS 2-D solute 'anspo model and a corresponding analytical colution. The appendices include a des iption d listing of M0CMOD84, description of several analytical solution techniques,,and a pro dure for estimating conservative groundwater velocity values. ,

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