Swift II Self-teaching Curriculum: Illustrative Problems for The … · 2012-11-18 · swift i...

SWIFT I SELF-TEACHING CURRICULUM:

ILLUSTRATIVE PROBLEMS FOR TE SANDIA WASTE-ISOLATION FLOW AND

TRANSPORT MODEL FOR FRACTURED MED EA

Hark Reeves*

David S. Ward*

DRFT - INFOWMAL AN PRELIA.NARY AN" .AS SUCh*AY OM RRMS OT YET COr RED. FOR#1SE PRIME STRBUON MD OT FORUORA RELaSE WIPOU CONSEN OF AUJTMOS

March 1984

Sandia National Laboratories

Albuquerque, New Mexico 87185

Operated by

Sandia Corporation

For the

U. S. Departmenr of Energy

Prepared for

Division of Waste Management

Office of Nuclear Material, Safety and Safeguards

U. S. Nuclear Regulatory Commission

Washington, D.C. 20555

Under Memorandum of Understanding DOE 40-550-75

NRC FIN No. A1166

8409120064 840228PDR WRES EXBANLA-1 166 PDR

. .

* GeoTrans, Inc.

DAFT INFOAMAL AND riL;., .. * ,, ; LY CONTAIN ERRo NOT YET CORRWCED, FOR

314t0 W PRIVATE DMBM OND NOT fOR ABSTRACTEX9LMREWWWI USOFAOMSE

!vveral documents have been written describing SWIFT II, the most urrent

version of the SWIFT Model. One, Reeves et al 1984a], describes the theory

and implementation, and another, Reeves et 81 [1984b), describes the data

input. Two others, Ward et al 11984a) and Ward et aL 11984bW deal with code

assessment through verification and field application and through

benchmarking, respectively. This document, however, has an entirely different

function. It, like the work of Finley and Reeves 19811 for an earlier

version, is devoted to assisting the analyst who desires to use the SWIFTCode. This code is quite general in terms of both the pocesses (flow, heat,

brine and radionuclide transport) and the media (single and/or dual porosity,

confined or unconfined) which it considers. Consequently a document such as

this one is necersary. Nine examples are presented to illustrate the use ofSWIFT II in doubly porous and unconfined aquifers. Each problem and its

numerical solution are described. Then several exercises are presented inorder to assist the reader with the input and the output and, occasionally,

with the interpretation of the results. For each case a printed input listin

and a microfiche output listing .e provided.

ii

tit

ACKNOWLEDGEMENTS

The authors would Like to acknowledge the work of 'Ka. Lynette S. Knippa

for her typing skills and for her determination to meet rther stringent

deadline. This document would not have been ossible without her help.

IV

v

TABLE OF CONTENTS

Section Page

1 INTRODUCTION .................... *e, .. q1-1

2 SOLUTE TRANSPORT THROUGH DUAL-POROSITY EDIA...................

2.1 PROBLEM 1. TRANSPORT OF A DECAYING RADIONUCLIDE IN A

FRACTURED POROUS MEDIA [TANG F.T AL, 1981).................

2.2 PROBLEM 2. TRANSPORT OF A DECAYING RADIONUCLIDE IN A

FRACTURED POROUS MEDIA HUYAKORN* 19831...................

2.3 PROBLEM 3. TRANSPORT OF A RADTONUCLIDE CHAIN IN A

FRACTURED POROUS MEDIA INTRACOIN, 19831...............*e

2-1

2-3

2-23

2-3S

3 iLUID FLOW THROUGH DUAL-POROSITY MEDIA ......................... ............, 3-l

3.1 PROBLEM 4. ANALYSIS OF WELL-TEST DATA FOR A DOLOMITE

FORMATION PAHWA AND BAXLEY, 19801....... 3-3

3.2 PROBLEM 5. ANALYSIS OF WELL-TEST DATA FOR THE MUSQUODOBOIT

AREA, NOVA SCOTIA (PINDER AND BREDEHOEFT, 19681........... 3-23

4 FLOW AND TRANSPORT THROUGH AN AQUIFER WITH CONFINING LAYERS.... 4-L

4.1 PROBLEM 6. DRAWDOWN FROM A FULLY PENETRATING WELL IN

A LEAKY AQUIFER KANTUSH, 19601 .................... vote*** 4-3

4.2 PROBLEM 7. HEAT TRANSPORT DURING FLUID INJECTION

[AYDONIN, 19641 ........ .............. 4-17

5 FLOW WITH A FREE-WATER SURFACE ........ ..................... 5-1

5.1 PROBLEM S. THE DUPUIT-FORCHHEIHER STEADY-STATE

PROBLEM [BEAR, 197Z! ...... . .... ... . .*.*. . *........ .. 5-3

5.2 PROBLEM 9. THE OUSSINESQ TANSIENT-STATE PROBLEM

(BEAR, 19721 . .... ........................ 5-15

NOTATION

REFERENCES

APPENDIX

................. .....................................

.................................. ...

CONVERSION OF INPUT DATA FROM SWIFT TO SWIFT II ..............................

6-1

7-1

8-I

VI

vii

LIST OF FIGURES

Figure Page

2.1-1 Problem . Schematic Diagram...... ..................... 2-4

2.1-2 Gridding of the Fracture ................................... . 2-9

2.1-3 Gridding of the Rock Matrix ........ ........ .. ....... . 2-10

2.1-4 Listing of the SWIFT II Input Data ............................. 2-11

2.1-5 Radionuclide Concentrations Within the Fracture for a

Prismatic Characterization of the Rock Matrix.................. 2-18

2.1-6 Radionuclide Concentrations Within the Rock Matrix for a

Prismatic Character.zation of the Rock Matrix .................. 2-19

2.2-1 Problem 2. Listing of SWIFT U Input Data.........*.......... 2-25

2.2-2 Radionuclide Concentrations Within the Fracture for a

Spherical Characterization of the Rock Matrix.... .............. 2-31

2.2-3 Radionuclide Concentrations Within the Rock Matrix for a

Spherical Characterization of the Rock Matrix .................. 2-32

2.3-1 Problem 3. Schematic Characterization of Transport Within the

Fracture and Diffusion Within the Matrix......... ............. 2-36

2.3-2 Listing of SWIFT II Input Data ................................. 2-41

2.3-3 Gridding of the Fracture/Matrix System......................... 2-50

3.1-1 Problem 4. Schematic Diagram of Well-Test HRA Within the

Magenta Dolomite Formation ........................ * ** *** .... *.... 3-4

3.1-2 Conceptual Models for the Secondary Fracture Porosity .......... 3-5

3.1-3 Listing of the SWIFT II Input Data . ..................... 3-11

3.1-4 Flow Rate and Drawdown of the H2A Slug Test Within the

Magenta Dolomite Formation* ......... a.... 69e.444*4. ... . s 3-18

3.2-1 Problem 5. Location Map of the Musquodoboit River Basin ....... 3-24

3.2-2 Geologic Map of usquodoboit arbour Area, Nova Scotia ......... 3-25

3.2-3 Geologic Cross Section, Hurquodoboit Harbour Area,

Nova Scotia . 4.. . 4. .... . .. . .. *. ... ... 3-26

vifl

LIST OF FGUMES

(Continued)

Figure . X

3.2-4 Location of Observation Wells and Characterization of

Hydraulic Properties... .... . ...... .. ....... * ... ...... 3-27

3.2-5 Observed and Simulated Drawdowns for Wells 1, 2 and 3 Using

a Homogeneous Dual-Forosity ModeL .............................. 3-28

3.2-6 Time-Drawdown Curves Obtained From the Simulation of Finder

and redehoeft 119681 ................... ........... *. ......... 3-29

3.2-7 Listing for the SWIFT 11 Input D.. 3-32

3.2-8 Simulation of the Observed rawdowns for Well I Using a

Homogeneous Dual-Porosity Model ....... ........ ... ............. 3-36

4.1-1 Problem 6. Schematic Diagram of a Fully Penetrating Constant-

Discharge Well in a Leaky Aquifer .. , . , . 4-4

4.1-2 Numerical Characterization of the Leaky-Aquifer System ......... 4-8

4.1-3 Listing of SWIFT It Input Data ............... ..... 4-9

4.1-4 Graphical Comparison of the Numerical Results from SWIFT and

the Analytical Solutions of Hantush for a Radial Distance

of 117.4 m......................................... 4-14

4.2-1 Problem 7. Radial Heat Transport Within an Aquifer with

Losses to the Over/Underburden .......... 4-18

4.2-2 Listing of SWIFT 1I Input Data................................ 4-22

4.2-3 Temperature Breakthrough Within the Aquifer at 3.5 from

the Injection Welle........... 4-27

5.1-1 Problem 8. Schematic Diagram for the upuit-Forchheimer

Prob5em ........... . . . . . . . . . . . . . .... . . . . . . . . . . .a . . a . . . 5-45.1-2 Listing of SWIFT I Input Data ......... 6.0....Go........ ...... 5-7

Ix

LIST OF FIGURES

(Continued)

Figure Page

5.1-3 Steady-State Free-Water Surface for the Dupuit-Forchheiser

Problem.,.,.*,.6 ... ......................................... 5-LI

5.1-4 Geometrical Characterization of the Gridding, Boundary

Conditions (Labelled "ALF") and the Recharge. .t............ 5-12

5.2-1 Problem 9. Schematic Diagram of the Boussineaq Problem...... . 5-16

5.2-2 Transient-State Free-Water Surface for the Bouasinesq

Problem........ .. .... * *....... a@*ee.0. * .... .* .......... 5-18

5.2-3 Listing of SWIFT It Input Data ................... .. 5-20

5.2-4 Geometrical Characterization of the Gridding and Boundary

Conditions (Labelled "ALF") in Relation to the Numerical

Solution..... ............. .. *. ................. *........ 5-26

x

.

xl

LIST OF TABLES

Table __

2.1-1 Problem . Input Specifications ............................... 2-(

2.3-1 Problem 3. Input Specifications ............................... 2-3'

2.3-2 Nuctide Inventory (12) nd Matrix Retardation ( 3) ............. 2-4(

2.3-3 Breakthrough Parameters.......... .................... ........ 2-4

2.3-4 Breakthrough Profile ............... ...... *..... .... 2-41

2.3-5 Material-Balance Summaries .......... ........................... 2-5

3.1-1 Problem 4. Input Specifications.............................. 3-1

3.1-2 Observed DecLne in Water-LeveL Height and Bottom-Role

Pressure........ ... *S ... 3-....3

3.2-1 Problem 5. Input Specifications........................ ....... 3-3

4.1-1 Problem 6. Input Specifications .......................... ... 4-

4.2-1 Problem 7. Input Specifications.............................. 4-2

5.1-1 Problem 8. Input Specificationa.............................,. 5-

5.2-1 Problem 9. Input Specifications............. ...... 5-1

xii

Wf - INFORMA AND PRELIMNARY MD AS SOANY o TAIN ERRS NOT YET MECED. FOlm4gou PfiVTE DItRBUO MD Not Vol

INT&ODU mP± rElfMASE W 1liu OSENT OF ALITMO.

The SWIFT Model (Sandia aste-Isolation Flow and Transport Model) has

been developed for the evaluation of repository-site performance. It i a

fully-coupled, transient, three-dimensional model, and it is implemented by a

finite-difference code, which solves the equations for flow, heat, brine and

radionuclide transport in geologic media. Having evolved from the U.S.

Geological Survey Code, SWIP (Survey Waste Injection Program (INTERCOHP,

19761), this code has experienced continuous improvements since 1977 Dillon

et al, 1978, and Reeves and Cranwell, 9811. SWIFT I marks a significant

extension of the development in that the model can now treat three additional

types of media in a cost-effective manner. Two are confined dual-porosity

systems and include a fractured porous material and an aquifer with conductive

confining beds. The other is an unconfined aquifer with a free-water

surface. As such, the model has become a very comprehensive and effective

tool for evaluating the processes envisioned for a repository of high-level

nuclear wastes. Furthermore, its applications extend beyond nuclear-waste

isolation to, for example, aquifer thermal-energy storage, liquid-waste

disposal by deep-well injertion and migration of contaminants from surface

disposal sites.

At the same time, however, the use of SWIFT does make rather heavy

demands on the analyst. He must have a familiarity not onLy with the basic

science but also with the mathematical model*, the numerical model*, the

code*, its input and its output. Current documentation (Reeves et al,

1984a,b) is available covering each of these items, evertheless, probably

the esist way to earn the SWIFT Model, or any other model, is through a

meaningful set of worked problems. Such is the basic premise for both this

report and its precursor, SWIFT Self-Teaching Curriculum Finley and Reeves,

19811. As a matter of fact, that earlier report is appropriate for SWIFT

II. A total of 11 problems are presented there in the input format used by

SWIFT, Release 4.81. However, with only minor changes, as described in the

* See Silling 119831 for a definition of these terms.

1-2

Appendix of this report, the input data sets listed there may be used to

illustrate the application of SWIFT II, Release 12.83, to single-porosity

media.

In this document 9 problems are presented. Fracture/matrix problems are

presented first, followed by aquifer/confining-bed and uncvnfined-aquifer

problems. In each case the discussion proceeds from a description of the

problem to a description of the numerical simulation. For convenience, a

definition of the symbols is reserved for Section 6. Further, the oft-

referenced documents of Reeves et a 11984a.bl are referenced simply as the

Theory and Implementation and the Data-Input Guide, respectively. Several

exercises are placed at the end of the discussion for each problem, which are

directed at two different levels of expertise. The more basic Level I

familiarizes the reader with the input, which is printed, and the output,

which appears in microfiche affixed to the back cover. For these exercises

the only necessary resources, other than this document itself, are the Data-

Input Guide and the Theory and Implementation. The more advanced Level 2

tests the reader's ability to extend or interpret the basic results. Here a

computer and the SWIFT I Code may also be necessary.

2-1

2 SOLUTE TRANSPORT THROUGH DUAL-POPOSITY MEDIA

2-2

2-3

2.1 PROBLEM 1. TRANSPORT OF A DECAYING RADIONUCLIDE IN A FRACTURED POROUS

MEDIA [TANG ET AL, 19811.

2.1.1 Objective

* To ilustrate fracture/matrix transport with a prismatic

characterization of the rock matrix.

2.1.2 Description of the Problem

Problem Statement. A thin rigid fracture is situated within a saturated

porous rock matrix as shown in Figure 2.1-1. Both fracture and matrix are

semi-infinite in their extent. Radionuclides, which derive from a source of

constant strength, are convected and dispersed through the fracture by a

constant velocity field and are diffused into the rock matrix.

Transport Equations. Mathematically the transport within the fractures

is governed by the equations

- a3X (ecu) (a D C) r - KOpC * (KPC) (2.-L)

Transport within the rock matrix is assumed to occur in a direction

perpendicular to the fracture and is governed by the equation

a (PD' -+ * 'p'C' a (K'*pOC') (2.1-2)

Coupling then arises through the flux at the fracture/matrix interface:

r .- *o'DI Tc (s,0) (2.1-3)

2-4

C=1.0SOURCE

I .

z

Iv

I

IXU-

8

ccK-

Figure 2.1-1. Problem 1. Schemacic Dlagram.

2-5

The composite parameters are defined here in terms of the basic parameter

set. They are the dispersion/diffusion of the fracture:

0 a Lu X (2.1-4a)

the diffusivity of the fractures:

D a 40* (2. t-4b)

and the diffusivity of the rock matrix:

D' a Do 'D* (2.L-4c)

The remainder of the parameters are defined in Section 6.

Initial/Boundary Conditions. Within the rock matrix the initial

concentration is zero:

C'(X,,t-0) -20 P s > x > 0 (2. L-5a)

and fracture and matrix concentrations are identical at the interface-

C'(X,8=0,t) C(x't) , t > (2.1-5b)

Within the fracture, the initial concentration is also zero:

C(xtw0) - 0 x > (2.1-6a)

and the boundary concentration is unity:

C(x-0,t) t > (2. 1-6b)

Input/Output Specifications. The input data for this problem, given in

Table 2.1-1, is the same as that prescribed by Tang et al for their lcw-

velocity case (p. 561 f). The desired output consists of two sets of spatial

2-6

Table 2.1-1. Problem 1. Input Specifications.

Parameter Symbol VaLue

Fracture Width 2d 10-4 m

Matrix Porosity 0.01

Matrix Tortuosity 0.1

Fracture Dispersivity CL 0.5 m

Molecular Diffusion in Water D* 1.6x1O-5 2

Kalf Life r 12.35 y

Decay Constant 0.056L y1

Matrix Retardation K' 1.0

Fracture Retardation K 1.0

Fracture Velocity v 0.0. M/d

Fracture Porosity 1.0

2-7

distributions. Curves of concentration versus distance within the fractureare called for at the values of time, t 100, 1000 and 10,000 d. Then asingle curve of concentration versus distance within the rock matrix is calledfor at the time t 10,000'd and the position x - 1.5 m.

2.1.3 Numerical Simulation

Discussion of Code Input. Two items are of interest here. Specifically

they are: (1) representing a semi-infinite domain with a finite domain and

(2) specifing the flow and diffusion within the systeot. For the first ofthese, the steady-state solution of Tang et al is most helpful. This solutionprovides one characteristic length for penetration into the fracture and

another for penetration into the rock matrix:

L 1 {(1l2)xL - ((1/4)aL2 + /D21- (2.1-7)and

a " (DtX)/2 (2.1-8)

where

D y + D* (2.1-9a)

D'- D* (2.1-9b)

* - x~ * *t (D'\))2/ (2.1-9c)

With values taken from Table 2.1-1, these expressions yield

L = 1.4 m (2.1-10)and

a - 0.30 m (2.1-11)

.

2-8

The domains of simulation are taken to be somewhat arger than these

characteristic lengths, i.e.#

L : 10.24 m (2.1-12)

for the fracture (Figure 2.1-2) and

a a 1.2 (2.1-13)

for the rock matrix (Figure 2.1-3a).

Spatial increments in the fracture and matrix domains are then taken to

be significantly smaller than he characteristic lengths. Nevertheless, some

numerical experimentation is necessary for gridding both the spatial and the

temporal domains. There is, however, one consistency check:

As (2D'At)/2 (2.1-14)

for the matrix, which is a useful guide. Figures 2.1-2 and 2.1-3 show

pictorially the gridding adopted for the two spatial regimes. Figure 2.1-4,

then exhibits this information as it i *pecified in the data input. As

shown, the globally connected mesh of SWIFT It is used to simulate the

composite fracture/matrx system with the individual increments specified in

Cards R-l (x), R-18 (Ay) and R1-19 (z). The one-dimensional mesh,

locally connected to only one global grid block, is then used tc simulate the

rock matrix, which is here mbedded within the global mesh. ridding, in this

case, is specified by the three parameter values provided in Cards ROD- n (i)

and RD-2-2 (a and As). Given the number of increments n and the

increment As at the fracture/matrix interface, the code generates the local

mesh automatically.

Continuing on to the second item, the flow within the global system is

maintained using injection and production wells in the end blocks of the

global system, he common rate being

q - 2dyv - 1.157 x 10 1m 3S , R2-6 (2.1-15)

f5 10'< .512 1.02

*12.05 3.07 9.22 10.24

I i / / I I2.4

I I / I I I0 2.5 x1O" 7.5 104' .768 1.54 2.56 9.73

I

Figure 21-2. Cridding of che raccure.

2-10

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He - , - - - I

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el I -- - -I -1 -1 000

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0

=

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jZ3kSE?qotZ567flt3LZ 34567sq012345670qGt2lt:56789012)ST$901Z3456?0lI23456I q0._________,_ _e _* _ _ _ _ _____ .___e_ _, ___. _ ____. _ _ ._________.

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2-16

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2-17

Since both fracture and matrix are included within the global blocks (IFD 0

in Card ROD-3), the global porosity is specified by

* = 4/(ad) a 4.167 x 10 5 , RlO (2.1-16)

This ylds, then, from Equation (2.1-4a), the value

D * 6.6672 x 1014 2I , R1-2 (2.1-17a)

for the diffusion of the global system. The corresponding relation for the

rock matrix is, from Equation (2.1-4b):

D' - 1.6 x 1012 M In (2.L-17b)m

Results. Figures 2.1-5 and 2.L-6 show the results obtained from the

SWIFT Code plotted alongside the analytic results of Tang et a. Both within

the fracture and within the matrix, the two show reasonable agreement.

Discussion of the Code Output. The numerical results exhibited in these

figures are presented in much more detail by the microfiche listing (inside

the back cover). At this point, the r 4 er is encouraged to scan the listing

for a general familarity, but with only one specific purpose, i.e., to locate

the data used in these figures. The tables which are denoted there by the

words "dual porosity" or "rock matrix" refer, in this case, to the rock matria

of a fracture/matrix problem. In Problems 6 and 7, however, these tables

acquire a new meaning in that they there refer to the confining beds of an

aquifer.

in order to examine the concentration distributions, the reader should

note specifically three tables. The first of these tables is entitled X-

Direction Distance to Grid-Block Center". It provides distances along the

fracture to the centers of the grid blocks, As shown, the distance x - 1.5

lies in Global Block 13. The second, "Dual-Porosity Block Numbers," then

identifies, with unique number, the local one-dimensional rock matrix units

which are imbedded in the global blocks. Note that such a unit is not

imbedded in Block 1. This block, as discussed under Problem 2, is used only

for the purpose of establishing a constant-concentracion boundary within the

fracture. Note also that Local Unit 12 is associated with Global Block 13 r

2-18

- ANALYTICAL

0.8 o SWIFT

Z 1 0 0 ~~000 d

< ~100 2.d. s *

z .40

0.2-

1.0 2.0 3.0 4.0 5.0DISTANCE DOWN FRACTURE ()

Figure 2.1-5. Radionuclide Concentrations Within the Fracture for a

Prismatic Characterization of the Rock Matrix.

- ANALYTICAL

0 SWIFT

ZiO A\ ~~~~~~~~~~~~t 10,000 d -:

i- .2 1OO

0 .2 A ,6 1.0 1.2

DISTANCE INTO MATRIX ()

Figure 2.1-6. Radionuclide Concentrations Uithin the Rock Matrix for a

Prismatic Characterization of the Rock Hatrix.

2-20

hence, with the distance x 1.5 n measured within the fracture. The third

table "Specific atrix Discretiration Parameters" gives nodal locations within

the rock-matrix units. Note that Node 12 is located at the fracturelmatrix

interface. Thus, for example, Node 4 is located at a distance of about 04

from that interface.

Tables of fracture concentrations, labelled "Component-1 Concentration,"

and matrix concentrations, labelled Component-I Radionuclide Concentration

Within the Rock Matrix," appear at selected time intervals, with the Last such

tables corresponding to 0,000 d. The calculated fracture concentration C a

0.318 for Global Block 13 x 1.5 ) does agree with Figure 2.15. Also, the

matrix concentration C' 0.068 for Local Unit 12 and Local BLock 4 (a 0.4 m

from interface) is consistent with Figure 2.1-6. Other values may be checked,

as desired.

2.1.4 Exercises

Level 1. Using the Input-Data Guide, the Theory and Implementation and

the microfiche listing of the output, as required, complete the following

exercises:

Exercise 1: From Table 2.1-1 and Equations (2.1-16) and (2.1-17), what data

are required for the blanks labelled "" through "5" in the

input-data set?

The control parameters called for here should specify steady-

state flow, the international SI system of units, transient-

state radionuclide transport and no convection within the rock

matrix (the local system). In this case convection within the

matrix is of negligible importance, and, for computational

efficiency, the proper control parameter should be set to

indicate "n.. convection".

Lxercise 2: From the physical properties specified in able 2.1-2, what are

the data required for the blanks latelled "6" through "8" in the

input-data set?

2-21

Exercise 3 Check the answers for Exercises and 2 by referring to the

output listing, specifically to the echo of the input data which

appears at the beginning.

Level 2. The reader who wants to develop a deeper understanding of the

code should also consider the following exercise*:

Exercise 4: A basic assumption of the foregoing analyses, both analytical

and numerical, is that diffusion in a direction parallel to the

fracture may be neglected. Evaluate this assumption for the

problem of Tang et a.

In most applications where fracture transport (convection and

dispersion) dominates over matrix diffusion, it is safe to say

that this assumption is valid. In some applications, however,

it may be necessary to check this assumption. One approach is

simultaneous discretization of both fracture and matrix, hich

involves the gridding of a two dimensional plane. Such a

simulation has been performed by one of the authors with the

basic assumption found to be valid.

* For the reader with no previous exposure to the SWIFT Code this exercise

will Likely be too difficult at this point and should be postponed.

2-22

2-23

2.2 PROBLEM 2. TRANSPORT OF A DECAYING RADIONUCLIDE IN A FRACTURED POROUS

MEDIA IHUYAKORN, 19831

2.2.1 Objective

* To illustrate fracture/matrix transport with a spherical

characterization of the rock matrix.


This problem is identical to Problem I with only two exceptions.

Firstly, it is assumed that the exposed surface area between fracture and

satrix, per unit fracture length, is greater here and may be more

realistically approximated by a spherical surface (Figure 2.1-3b) for the rock

matrix. Thus, Equation (2.1-2) of the previous section is replaced by

I a (sp'D' aC + r - K'$0c' C a (OpICI) (2.2-1)s; s m 3 a

The radius for the spherical units is chosen to be identical to that used in

Problem 1 for the length of the prismatic units:

a - 1.2 m (2.2-2)

Secondly, the matrix diffusion is reduced by a factor of approximately

three relative to that of Equation (2.1-17):

D' a 5.787 x 10 3 m2s (2.2-3)m

The desired output, as in the previous problem, consists of two sets of

spatial distributions. Rowever, the values of time are changed to t 441,

3619 and 90,615 d for the distributions within the fracture and the rock

matrix. The position for the latter is also canged to x - 1.0 m.

2-24


Discussion of the Code Input. To simulate the semi-infinite fracture,

two boundary conditions are specified here, just as they are for Problem 1.

At the maximum extent of the system (10.24 ), a convection-only condition

3T (x-L) 0 0 (2.2-1)

is easily specified in the code input (Figure 2.2-1) by default.

At the origin, however, the specification of unit concentration (Equation

2.1-6b) is considerably more complicated. Since SWIFT is designed to evaluate

repository-site performance, only facilities for radionuclide sources have

been provided. owever, a constant-concentration boundary condition may still

be used by taking the boundary block (Block 1, in this case) to be an

essentially infinite well-stirred reservoir. In fact, this is a basic

physical definition for a constant-concentration condition. Distances, of

course, are measured by taking x - 0 at the interface between Blocks I and 2,

and the dual-porosity region begins with Block 2 (see Card ROD-3). To

implement the concept of a well-stirred reservoir, four separate steps are

taken here.

Firstly, the thickness of lock I is increased to

AZ (1.91985 x 10 + .4) , RI-26 and R-19 (2.2-2a)

using the modification cards. Secondly, the concentration of this block is

specified as

Cl ' 1 1-4 (2.2-2b)

Thirdly, provision is made to replace, with a source, the mass lost from the

reservoir by both decay and convection:

q - C p()4ax 6yhz q) 7.12 x 10 kg/s , R2-LO (2.2-2c)

2-25

1 2 q 5 & 7 6L13�.5agq01Z34',6t6q01Z3456Tqq0IZ3i54?5qOIZ34.567Sq0Li345676q01Z3456 leqoKzli5Etoqo

* -- * *-0.-*-. 0.S* *K

I KPR05LEI� P40. 2. THIEE-COMPOP4EffT #4UCLIOE TRANSPORT KN PRAeUREO RECtA.K

2 KRZ�83. CDI4PARISI3N suits 'FTRANS' - TPHEA1CAL IEPREIVI1TATIQNI

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*5-3-2

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l I I I I 0

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10-1

10-2-1

ERENCED IN TEXTERENCED IN EXERCISE * 10-2-2q... eeg*. .... e.... see

10-2-2

ROD-i

100-2

100-3

t0fl-3-SLt4

1.0 11-1

0.50 0.0 1.00E�50RL*2

1l-Z.5-1

11.2 .5-2

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A 1-2.5-2S.....

1000.0 11-3S..,.'

11-4

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li-I

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-- *f**n *.flSn*. *e�a*

Figure 2.2-t. Problem 2. tLsting of SUIFT 11 Input Data.

2-26

0RAr

; 1 2 3 * 5 7 T a1i3*SbflQ4lR4Sl7S01Z34iSi7l6f11lQ0i13*51a76,0190137901fl45l76901Z3*sb?690

,_______ ,_______ ._______._a______._n______- - *--- -e----- -

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IS � S-s-a*----a---�n----4n--aa


a

2-27

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Fi6ure 2.2-l Concinued.

2-28 OA

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* * *… * ... i n *.Sb...S * - e * . *~~~~~~~~~~~~~~~~~~~~~~~

Figure 2.2-1 Cont inued.

2-29

Okr

I 2 3 . S T a123*41 7*qO IZ34S67SC01214 6tq9 1 IS9w I* *O . a234 S676q01ZI45679012 34 S6

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2-30

Here the various component terms, except for C and Ai, which are given

above, are prescribed in Cards R-6 (p), RO-1 (T Ln(2)/)), R1-20 (),

RI-l7 (AxI), R-18 (AyI) and R2-6 (q).

Fourthly, and lastly, to implement the concept of a !'ell-stirred"

reservoir, the dispersivity

aL 4 10 u , R1-2.5-2 (2.2-2d)

is selected for Block I only. This value, without question, is sufficiently

large that the transmissibility (Theory and Implementation, Section 6.1.3)

cannectir. Blocks I and 2 reduces to a boundary-like transmissibility for

Block 2. In te data input, this value is given to Rock-Type 2. Rock-Type 2

is then ascribed in RIA-1 to Block I only. Rock-Type 1,

with aL = 0.5 m, refers to the transporting region of the fracture, Blocks

2 - 21.

Results. Figures 2.2-2 and 2.2-3 show the results from the finite-

dirference SWIFT plotted alongside those of uyakorn's finite-element code

FTRANS. Both within the fracture and within the matrix, the two show

reasonable agreement. The dissimilarity between Figure 2.2-2 and Figure

2.1-5, however, is somewhat striking. Obviously, transport within the

fracture is significantly more retarded by the spherical-matrix

characterization than by the prismatic-matrix characterization. In spite of

reduction of approximately three in the matrix diffusivity, the spherical

characterization still yields a retardation which is greater, by a factor of

three or more, than that for the prismatic characterization.

Discussion of the Code Output. The results shown are dependent, at teat

in part, upon the convective velocity and upon the constant-concentration

condition discussed earlier. To check these two, the reader is referred to

the microfiche listing. Insofar as the fluid velocity is concerned, note in

the table "Homogeneous Reservcir" the value of porosity for the composite

global system ( - 4.167 10 ). Note also, in the first recurrent series

output, the first table, which is entitLed "INDQ IWELL .. T". The purpose

the computations referred to here is simply to establish a steady-state flow

with boundary wells, Well rates are given and the printing of the Darcy

2-31

1.0 -

-FTRANS

0.8 0 SWIFT

90.615 d

z° 0.6 3619 d

cc ~~441 dzw

0 \

0.20

0 1.0 2.0 ;.0 4.0 5.0

DISTANCE DOWN FRACTURE (m)

Figure 2.2-2. Radionuclide Concentracions Within the Fracture for a

Spherical Characterization of the Rock Matrix.

2-32

0.4

- FTRANS

0.3 90,615 d o SWIFT

z x = 1.0m0 ~~~3619 d

0.2 41 d.2z

0

0.1

0 0.2 0.4 0.6 0.8 1.0

DISTANCE INTO MATRIX (m)

Figure 2.2-3. Radionuclide Concentracions Within he Rock atrix for a

Spherical Characterization of the Rock Matrix.

2-33

velocity is prescribed by the control parameter IIPRT 13. The table "X-

Direction Darcy Velocity" then provides the desired check on the convective

velocities. As shown in the table, it is constant throughout the entire

length of the ystem, and it has the magnitude

v u (4.8208 x o 12 /4.167 x 10 ) w/s - 0.01 m/d (2.2-3)

which is consistent with the problem specifications (Table '.l-0.

To verify the constant-concentration boundary condition, it is sufficient

to examine the radionuclide concentration tables labelled "Component-L

Concentration." As the reader will note, there are seveiral such tables, which

are displayed at selected time intervals out to the MnximuM time, t 90,615

d. In each case, the C 1.000 condition is maidrAined to at least 4

significant figures.

2.2.4 Exercises

Level 1. Using the Input-Data Guide, Theory and Implementation and the

microfiche listing of the output, as required, complete the following

exercises:

Exercise I

Exercise 2

From the description of the problem, Equation (2.2-3) and Table

2,1-1 what physical parameters are required for the blanks

labelled "1" through 4" in the input-data set (Figure 2.2-1)?

Check the answers by referring to the echo of the input data

printed at the beginning of the output.

Interpret the output-control parameters identified in the last

R2-13 Card of Figure 2.2-1.


code should also consider the following exercise:

2-34

Exercise 3 If the retardation of the rock matrix were increased from ' -

to K' a 3 would the penetration of the radionuclide front in

the fracture b increased or decreased? To answer this

question, the problem may be run anew with the value of he

distribution coefficient on Card RID-4 change.d to kd

7.41 x 10'6 kg/m3 .

2-35

2.3 PROBLEM 3. TRAKSPORT OF A RAD1ONUCLtDE CAIN IN A FRACTURED POROUS

MEDIA UINTRACOIN, 19831

2.3.1 Objective

* To illustrate the coupled effects of fracture/matrix transport and

chain decay.


Problem Statement. Radionuclides, it is assumed, are buried in a

fractured medium. As indicated in the schematic drawing of Figure 2.3-1, the

system consists of parallel horizontal fractures with apertures 2d and

spacings 2a. It is infinite in its lateral extent but confined to a vertical

thickness b - At by impermeable beds. The radionuclides each from their

storage leach duration, 2) and contaminate the entire unit. They are then

transported laterally by a one-dimensional flow field. Convection and

dispersion occur within the fractures while diffusion and sorption occur

within the rock matrix. The geometry presented for the diffusion process may

"e characterized by one-dimensional prismatic units.

Transport Equations. Mathematically, transport within the fractures for

radionuclide r i governed by the equation

a 3 * aCG. (pe ) ax (PD ax) r (2.3-1)

+k A K #pC - A K~ -a(K toC)rr-l r-l r-l r-l r rPCr ' at r r

and transport within the rock matrix is governed by the equation

a Sc,S.(p D ) r (2.3-2)

k X K -1~ A K'p*C'1 La Ko~~r,r-l r-1 r-1 p r-1 r Arr'P' at rK Cr

Coupling then arises from the flux at the fracture/matrix interface:

AC'rr %0P'Dt -s (0) (2.3-3)r m ~as

PA -IN W.? *.:

N~~~~~~~at-'. ~ ~ ~ ~ ~ ~ ~ ~ ~ at.~

K.'

Figure 2.3-1. Problem 3. Schematic Characterization of Transport Within the

Fracture and Diffusion Within the Matrix~.

2-37

Initial/Boundary Conditions. Within the rock matrix the concentration is

initially zero:

C'(xG.tuO) 0 0( a a , > (2.3-4a)

a no-flux condition holds at the symmetry boundary:

aceSr- (x,6-at) f 0 , C 0 Q (2.3-4b)

and fracture and matrix concentrations are identical at the fracture/matrix

interface:

x(,s0,t) - Cr(xt) P t (2.3-4c)

Within the fractures the initial concentration is also zero:

Cr(xth0) - 0 a x (2.3-5)

and the infinite boundary is held at this initial concentration.

Source Specification. At x 0 however, the boundary condition for the

fractures must take into account decay/production processes within the

inventory and the leaching of the inventory. To do this, the time-dependent

source concentrations C are defined by the Bateman equations:r

dC

dc rr-l r-l r-l r r(2.3-6a)

with initial conditions

C (t-0) * I r/UT (Z.3-6b)

The B2 boundary condition chosen here is then expressed in terms of these

source concentrations:

UCr r x uCr , x (2.3-7)

0

2-38

Input Specifications. The geometry, the leaching process and the

transport process are characterized by the parameter values given in Table

2.3-1. Radionuclide-dependent processes are then cnaracterized y those given

in Table 2.3-2.


Discussion of Code Input. Within the input-data listing of Figure 2.3-2,

two items are selected for emphasis in this section. They are the fracture

velocity v and the boundary specification B2 . In contrast to Problems I and

2, the velocity here is established by means of a pressure boundary condition:

v a (Kfp(gfg )0I(p-pl )/L - 500 /yr (2.3-8)

In this equation the porosity of the global system

f = d/(ad) - 1.99996 x 10 5 RI-20 (2.3-9a)

is fixed by the input specifications (Table 2.3-1). The system Length

L 2,500 RI-17 (2.3-9b)

(the sum of alL increments in Card RI-17) is taken to be Large compared to the

observation length (500 m) in order to simulate a theoretically semi-infinite

system. Of course, the gravitational constant (g 9.81 m/s 2) is fixed

internally by the code.

Aside from these, all other parameters in Equation (2.3-8) are

arbitrarily specified to yield the desired interstitial velocity:

p - 1000 kg/t 3 R1-3 (2.3-lOa)

K - 1.99996 x 10 5 MIs RI-ZO (2.3-lOb)

Po = 388.44495 Pa RI-28-2 (2.3-1Oc)

and

PI ' RI-28-Z (2.3-10d)

2-39

Table 2.3-1. Problem 3. Input Secifications.- ~~ ~~~~~~~~~~. -

Parameter Symbol Value

Observation Length L1 500 m

Aquifer Thickness b 5.0 e

Fracture Aperture 2d 1.G x 10- 4,

Fracture Spacing 2a 5.O X

Fracture Porosity 4. 2.0 x 10-5

Fracture Velocity v 500 mly

Fracture Dispersity aL 50 m

Matrix Porosity of 5.0 x lo-

Rock Density PR 2700 kg/m3

Matrix Diffusivity D. l o1 2m2 t

Leach Duration 10 y

Table 2.3-2. Nuclide Inventory ( ad Matrixl Retardation (R3i.

Number Isotope Inventory Half-Life2 Retardation Distribution Specific

(Ci) (kg) t(y) K Coeff iciend Activity

k (a.3 11:g) A (Cilkg)

l 2 t, 24cm 0.7 4.074xl0-3 8.500x103 570 0.211 171.8

2 2 3 7Np 1.0 1.418 2.140x106 80 2.96xlO0 0.7055

3 0.OC4 4.l47x10-4 1.592x106 30 1.11X10 2 9.646

I No sorption occurs within the fracture.

2 The corresponding decay constant is given by - n(2)/T.

3 The two sorption parameters are related via K p k

2-41 D/4p

I I 3 * s 7 r Sall54?R0901245&7O9 012 34S7qO3I4I,6790Z34#blqoZ1i 567 qO1234S76901Z4S7&Tq0

._�__�_��__�4�___-- - - -___ _ __ --------- *--------- --- *--

I t II fraciLE4 No 3 INr*&COIN LEVEL CSE S 2.13,G2vPZ.TZEtLL 1-1-1 I

I I2 112153s ONE-0 TRAtSPORT IN FRACTURED EDIA WITH CHAIN ECAY R-1-2 t

3

10

5

6

71

a

9

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la

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12

13

14

15

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Roo-IL

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100-3

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kl-6

t1-10

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0.00t

0*001

*______ - -_ _--- _- - _ _-_- -_-__- -_-_-,_-_-_-- -_ _ -_-_ -

Figure 2.3-2. Listing of SWIFT 11 Input Data.

2-42 O

I 2 3 4 S a 7 6L21 7s,0sLZ3 so7e zqoZ s 67s0oL 3 sT0 I z3s6sqoIz36s6 lso0234s6e,023seqo

._______. ________._.__ _. ---------_ _-------_ _

26

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$ 0

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* 30.0 40.0 50.0 60.0I1 1.0tI 5.0001

I I .999q t -51 .qqqIE-51. qqq6f-S 1.q9tq6{-S 0.0

I

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I. 19 *, 10) 0* III) *I**v***o**.4..044*-.0....eoI1I 0 0 0I1 00I* 0I* *- e -4 - a n . . e . ... n. ae

Figure 2.3-2 Cont nued.

2-43 £W/Ar

51

52

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1 2 3 S 6 ? a12 34S6?GqOt1Z34S7?gqOIqa127gqto123456a1Z34S6790tZ3)56XqJ01231567G 01Z3567690

. __ _ _. _ * _ ____ . ______ _ * _ _ _ _ _ _ -_ . _ __

I II 0.0 0.0 0.19q96 0.0 tlA-a II I

too In.0 L.o tIA-S II II I I I I I I RlA-& II

54 1

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sq

60

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2-44

1 2 3 4 5 4 7 512)0s67sq623% s 9gsolrs67801 sol 6023 42~s~SO 6'z34s67ssoz23,stes78ol2345s7es0._ _ _ _ _ - --- - _- - - ------

I76 I5.67a46EI2 3.L53beI0

I77 I I -t -t -1

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0 0 0

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*-e *n *


2-45

In order to approximate the B2 boundary condition, Grid Bock is

specified as the source in Card RlA-6. This block, of course, is of finite

width with

Axt a 5 X , Ri-17 (2.3-hla)

However, this increment is only one percent of the observation length and

therefore should be an acceptable approximation to the theoretical boundary

condition of Equation (2.3-7), which is rescribed for an infinitesimal

increment.

Further, the source rate is specified within SIFT by the equation

Rwr ' -m rt )/wta (2.3-1 lb)

(Section 3.4 of Theory and Implementation) where

Rwr a source rate of radionuclide r kgrn3(bulk)/si,

Or a waste density of radionuclide r [kg/m3(waste)I,

V = volumetric density of wastes 1m3(waste)/m3(bulk)], and

ta a T = ach duration 1 1

Quantities ta, m(t-0) and pw must be specified in the input.

Prescribing the leach duration is straightforward. However, the other two

require some explanation. Here it is assumed arbitrarily that the waste

inventory occupies one m3 of volume so that

PW I xyz - 0.199996 , RIA-4 (0.3-12)

where Ax, I m, y 1 m and az * 5.0001 are specified in Cards RI-17,

RI-I8 and R-19, respectively. Thus, the initial densities mr(t-0) have the

same numeri,.al values as the nuclide inventories r given in Table 2.3-2, an.

the code internally solves for the Bateman decay (Equation 23-6a).

2-46

Results. Results are presented in the tabular form prescribed by the

INTRACOIN project, and they exhibit reasonable agreement with other codes used

in that study. Breakthrough curves at the observation point L 500 are

given, in abbreviated form, in Table 2.3-3 and, in expanded form, in Table

2.3-4.

D:scussion of Code Output. The concentrations given in these tables (in

Ci/m3) may be obtained from the computed concentrations (in kg/kg) via the

fluid density (1000 kg/m3) and the specific activities of Table 2.3-2. For

example, consider the concentration for 233U at 180,000 y. The point of

breakthrough is located in Block 102, which way be verified by examining, in

the microfiche output listing, the echo of the block sizes (see also Figure

2.'-3). The reader will also note in the echo of Card -l that Block 102 is

designated as a nuclide monitor block. This facility allows one to store (on

Tape 9) computed concentrations at this particular location. Here such

concentrations were stored so that they might be post-processed and Tables

2.3-3 and 2.3-4 written automatically in the INTRACOIN format. The

concentration computed for 23 3 at 180,000 y is printed in next-to-the last

table of the output listing, the one entitled 'Component-3 Concentration."

The value printed there as a mass fraction may be then converted to the

desired units:

C(Ci/ ) pA C(kg/kg) a 1.95x105 Ci/M3 (2.3-13)

using Tables .3-L and 2.3-2.

To provide a check on the calculation of concentrations such a these,

four material-balance summaries are printed at selected time steps. Two of

them, the one for unleached components and the one for leached, but

undissolved, components pertain to the source block. The other two tables.

one for dissolved and sorbed material in the fracture and one for such

material residing in the rock matrix, pertain to the system as a whole. Each

monitors the conservation of mass within a particular phase or subsystem and

then summarizes the information with balance quotients. Ideally these

quotients should equal unity. The various categories used in these tables are

defined in Table 2.3-5.

2-47

Table 2.3-3. Breakthrough Parameters.

NUCLIE C-SAK T-MX T+ (50t) T- (50%)(CI/CUA.M) (YEAR) (YEAR) (YEAR)

1 CM 245 1.6952E-11 49001. 28802. 79362.

2 NP 237 1.6665E-06 l5Booo. 77616. 464564.

3 U 233 1.9524E-06 18000 . 8245°. 688312.

…---.-------- _-----------------.-----._-_ ---------------

A

2-48

Table 2.3-4. Breakthrough Profile.

-------------- e- f ---- a--------------------------

TIKE(YEARS)

CONCENTRATION (CI/CU.K)____--- ---- _ _---- ------ ---- e_ -------

Cii 245 NP 237 U 233_-_a--a - a-------a------ --------- a eec -

3.50E+037.50E+031.15 E+O4I.55E+04I .95E+042.35E+O42.75c+043. 5E+043.55E+O03.95E+044.35E+O04.75E+0L5.15E+045.55E+045.95E+O46.35E+046.75E+047.15E+047.55E+047.95E+048.70E+O09 .50E+041.03E+051.11-+051. 19E+051 .27E+051 .35E+051.43E4051.51E+051.59E+051 .67E+05I . 5E+051 .86E+052.O4E+052. 36E+052 .68E+053.00EE+053. 32E+053 .64E+053.96E+05

5.1824E-t71.4297E-141.9872E-139.131OE-132.4360E-124.7405E-127.5312E-121.0400E-Il1.2972E- I1.4981E-111.6295E-1II .6698E- Il1.6855E-tl1.6280E- II1.5306E-1I1.4064E-111.267OE- I1.122tE-l1I9.7906E-128.4309E-126.1792E-124.276SE-122.8.703E-121.8777E-121.2019E-127.541 41Eal34.6455E-132.8133E-131.6778E-139.869lE-1I5.7358C-143.2981E-141 .5172E-144.1178E-153.7330E-163.1725E-172.5832E-182.0417E-191 .5793E-201.2023E-21

6.6839E-1 )1. 8597E-098.6g84E-ogt.2451E-084.3552E-087.1697E-081.0627E-07IA.654E-071 .9177E-072.4129E-072.944SE-073.5077E-074.0967E-074.7077E-075.3367E-075.9006E-076.6364E-077.3016E-077.9739E-078.6519E-079.9323E-071. 1302E-061.266BE-061.3952E-061.4974E-o61.5705E-06i.6192E-061.6485E-o61.6631E-061 .6664E-061.6611E-061.6494E-o6I.6256E-o6I.5743E-06I.660E-06

1.3541E-O01.2473E-061.1491E-061.0599t-O69.7947E-07

S. 2623E-II1. 1426E-095.2641E-091.4021E-082.8395E-084.8897E-087.5724E-081 .0886E-071.4819E-071.9347S-072.4444E-O73,0079E-073.6221E-074.2841E-O74.9906E-075.'7386E-076.5251E-077.375E-078.2028E-079 .o888E-071.o824E-061.2767E-061.478OE-O61 .6422E-061.7481E-061.8183E-O61 .8665E-O01 .9oooE-061 .9231E-o61.93B3E-061 .9I75E-o6I .9518E-061.9514E-061.9388E-06I.8923E-o61.8281 E-061.7551E-O61.6785E-06I 6013E-061.5255E-06

2-49

Table 2.3-4 Cont nued.

--- - n- -- --- --- --- -- -- -- -- ft --- - -

TIME(YEARS)

CONCENTRATION (Cf/CU.K)---------------------- ____---------

CA 25 NP 237 U 233-- ___________________________________________…__

4 .28E+054.60E+o54.92E+055. 24E+55.56E+055.88E+056.ZOE+056 .52E+056.84E+057. 6E+057.148E+057 .8OE+5

9.0552E-236.8074E-245.5825E-258.5281E-265 .0058E-263. 4527E-263 .5234E-268.2965E-266.8849E-z65. 1387E-263.9422E-261 .3475E-25

9 .0709E-078.4 9I E-077.8321E-077 .3009E-076.8206E-076. 38I43E-O75 .9872E-075 .6247E-075.2931 E-07X .9889E-07X..7092E-074 .4516E-07

I .4520E-06I . 38 I6E-o61 .3144E-061 .2509E-061. 1907E-061 .1339E-061 .0803E-o6I.0299E-069 .8243E-079. 376eE-078 .9559E-078. 559SE-07

.______________________.__..__,___________________

. 0

500

07

1 2 A

BLOCK

a x(m)

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2-51

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9

2-52

2.3.4 Exercises

Level L. Using the Input-Data Cuide, the Theory and mplementat.on and

4he microfiche listing of the output, as required, complete the following

exercises:

Exercise I Fill in the blanks lbelled "I" through "15" in Figure 2.3-1,

making reference as necessary to Tables 2.3-1 and 2.3-2. The

data called for here are either source- or radionuclide-

dependent information and may be verified b reference to the

output listing.

Level . The reader who wants to develop a deeper unierstanding of the

code should also consider the following exercises:

Exercise 2

Exercise 3

What effect does the rock matrix have on the transport?

Contrast the computed breakthrough-time parameters (Table 2.3-3)

with those for the case of convective transport in the fracture

with no matrix effects.

Table 2.3-3 indicates that the peak for 237Np occurs before that

for 233U. Why? Since the latter is in secular equilibrium with

the former, one would expect the peaks to occur at approximatley

the same time providing Chat the retardations were equal. Here,

however, 37Np has the larger retardation (Table 2.3-2). Thus,

should not 237Np peak later than does 233U?

3-1

3 FLUID FLOW THROUGH DUAL-POROSITY MEDIA

3-2

3-3

3.1 PROBLEM 4. ANALYSIS OF WELL-TEST DATA FOR A DOLOMITE FORMATION

[PAHWA AND BAXLEY, 19801

3.1.1 Ob jective

* To illustrate the effects of stress-relief fracturing upon a slug-

injectior. test.


Background. A number of slug and recovery tests were conducted by Sandia

Rational Laboratories and by the U.S. Geological Survey (ercer and Orr, 979;

Dennehy and avis, 19811 during the site-charactericacion work for the Waste-

Isolation Pilot Plant (WIPP) [Powers et al, 19781. Two formations of

particular interest were the Culebra and Magenta olmite members of the

Rustler unit, both of which are signifleant water-bearing rocks lying above

the Salado salt layer. These tests were analyzed by Dennehy and Davis using

an analytical model and, since many of the tests exhibited an anomalously high

flow rate at small values of time, they were analyzed also by Pahwa and Baxley

119801 using a dual orosity implementation of their numerical code.

In the SWIFT Self-Teaching Curriculum, Problem 11 Finley and Reeves,

19811, the R2A test (Figure 3.1-1) of the Magenta member was chosen, and the

conceptual model of Pahwa and Baxley was used. This model assumes that, in

addition to the primary porosity, which may itself represent connected

fractures extending throughout the dolomite, there are secondary-porosity

fractures which extend radially outward an average distance of 2000 ft (Figure

3.1-2a). It is then these fractures which affect the small-time response of

the flow rate. As in the work of Pahwa and Baxley, heterogeneity was used in

Problem 11 to characterize the secondary porosity.

Problem Statement. Here, a different conceptual model is invoked. It is

assumed that the secondary porosity is provided by stress-relief fracturing

which extends to a radial distance of about one foot surrounding the wellbore

(see Figure 3.1-2b). The generalized dual-porosity approach of the SWIFT I

Model is then used to simulate the measured data.

3-4

INITIAL WATER LEVEL

- STATIC WATER LEVEL

TUBING DIA. = 1.995 in.

CASING DIA. = 6 o

I

Figure 3.1-1. Problem 4. Schematic Diagram of Well-Test H2A Within the

Magenta Dolomite Formation.

3-4

_

E zwtT i �' 2 1., 45 ,-,. � jWTP�Rf,--:,.' , �,.' 4 ; : :I " " I ii,

- . f INITIAL WATER LEVELj

I

5(

i

8'

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57,'

_4I TUBING DIA. = 1.995 In.

*- CASING DIA. =6

_ _,________________ l, - , - , _ . - - - S x .r - U- I -, , , , , , o

,f , .. ..

*7774 W t - / - / - > J

DOLOMITE g -A, leAf z z - F F --

,f . . - -.1 . ..1- z v-' 'SP ' 4 7 *!~ 7 7

, 4 >- 17 ,, / ^- z / f

5'

1 F1 F , * 0~ Z F0 z .01 z le it_

Figure 3.1-1. Problemu 4. Schematic Diagram of VeLt-Tesc H2A W~ithin the

Magenta Dolomite Formation.

3-5

(a) Partially Penetrating Fracture(s).

(b) Stress-Relief Fractures

Figure 3. 1-2. Conceptual Models for the Secondary Fracture Porosity.

3-6

Flow Equations. athematically, within the primary porosity, the flow is

characterized by the equation

T r ar ) - bw Sat * r r a'7

(3.1-1)

and, within the secondary porosity, by the equation

,as2-S 'sat O < s a , rw r r0 (3.1-2)

where a is the average length of the connected secondary fracturing and ro

denotes the maximum extent of the stress-relief fracturing surrounding the

wellbore (radius, r). The coupling between primary and secondary porosity is

provided by the reLation

r '- AVK' a.g (5-O) (3. 1-3)

Initial/Boundary Conditions. Within the secondary porosity, the initial

drawdown is zero:

s'(r,s,t-0) - 0 O s a rw c r r0 (3.1.4a)

the drawdowns are set equal at the interfaces:

6'(r,s-O,t) s(rt) I r. r r0 (3.1-4b)

snd a symmetry condition is prescribed within the interior:

as, (,sna,t) 0 rw c r r (3.1-4c)

Within the primary porosity, the initial drawdown is also taken as zero

s(r,t=O) 0 r r (3.1-5)

.

3-7

and the water-level above the bottom-hole condition i escribed at the

vellbore radius r:

strwt) - s0(t) (3.1-5b)

Input/Output Specifications, Rather than the draudown formulation, the

SWIFT Model uses a pressure formulation. Thus, in the equations above,

drawdown is replaced by pressure as the dependent variable;

a Po P (3.1-6)

where p is the initial pressure. Also the storativity and transmissivities

are replaced by the equivalent expressions:

S a P(g/g c)(eW+c) (3.1-e)

T - Kb (3.1-7b)

and

Si a p(g/g )W'(c ') (3.l-7c)5 c v

Input data appropriate for both formulations are then given in Table

3.1-1. For the primary porosity, the values given there are the same as those

used by Pahwa and Baxley. For the secondary porosity, however, hich is

assumed herein co arise from stress-reLief fracturing, the values given are

entirely different from those of Pahva and Baxely and were developed

especially for this problem by an optimization procedure. The remainder of

the input data, i.e., the time-dependent height of the water Level within the

wellbore, is provided by Table 3.1-2. Actually, the analytical work of

Denneh. and Davis and the numerical work of Phwa and Baxley treated the

wellhisr sorage implicitly so that the height of the water level, in addition

to the injected flow rates, was calculated. We, too, have a special-purpose

update which permits such a computation. owever, in rder to use the

publically available code on his, an illustrative example, the measured wateL

levels are prescribed as input data, and only the injected flow rates are

cal culated.

3-8

Table 3.1-1. ProbLem 4. Input Specifications.

Parameter Symool Value

Density of Water p 62.4 ib/ft 3

Aquifer Thickness b 25.0 ft

Compressibility of Water c, 3.0 x 1o 6 i-

Primary-Porosity Media:

Porosity O.10

Compressibility of Rock CR 4.0 x 10-6 psi-

Hydraulic Conductivity K 2.0 ;: 104 ft/d

Storativity S 7.58 x 106

Transmissivity T 0.005 t 2 /d

Secondary-Porosity Media:

Porosity t 0.10

Compressibility of Rock cR 4. 10-5 i-Hydraulic Conductivity K' 2; .0 ft/d

Length a *. ft

Specific Storativity ' 1.86 x 10-6 ft-

3-9

TabLe 3.1-2. Observed Decliine in Water-Level Height and

Bottom-Hole Pressure.

Time Interval Water Level Bottom-Hole Pressure

(days) Height (ft) (psi)

IQ-4- lo-2 505 219

10-2 _ 101 496 215

0.1 - 0.2 485 210

0.2 - 0.5 468 203

0.5 - 0.8 450 195

0.8 - 1.0 434 188

1.0- 1.5 415 180

1.5 2.0 392 170

2.0 - 2.5 374 162

2.5 - 3.0 360 156

3-10


Diucussion of the Code Input. In contrast to the problems presented in

the previous chapter, this problem, like several that follow, employs a radial

coordinate system. The inner boundary condition derives from the presence of

a well of radius rw 0.276 t (Card R-22, Figure 3.1-3). This well

comrl..;ates with the aquifer through a weLLskin, which, in general,

represents a disturbed zone of modified hydraulic properties. For the SWIFT

Model it is assumed that the wellskin has no storage and that its hydraulic-

conduction property is characterized by a well index (Theory and

Imple entation, Section 4.1):

WI0 2v K3Az/tnIrIr) (3.1-8)

where quantities rl 1.0 ft, the position of the first node within the mesh.

and A .5 Et appear in Cards R-22 and R-23.

Here, the disturbed zone, e region assumed to have stress-relief

fractures, is also rpresented by a one-dimensional unit attached to the first

grid block (Card ROD-3). This unit does have some storage and a relatively

large hydraulic conductivity. In a sense, then, in this case the well index

(Card R2-7-.I) has been superceded by the one-dimensional unit.

Th? outer boundary is chosen consistently with the aximum simulation

time:

re (Tt IS) 100 ft (31-9a)

with r (Card R-22) being greater than the characteristic length r

r ' 2000 ft (3.1-9b)e

At this boundary, the numerical solution is matched onto an analytic

continuation for an (assumed) infinite external aquifer using the method of

Carter and Tracy [19601. The data appropriate for this method appear on Card

Rl-27 hrough R1-33. They should be examined carefull) since, in the

application of SWIFT, they are used frequently (Theory anJ Implementation,

Section 4.1).

3-11

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3-12 IDRAP

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3-17

Results. Figure 3.1-4 shows the simulated flow rates, which agree

reasonably well with the measured results. As indicated earlier, however, the

simulated values for times less than about 10-2 d were fitted to the measured

values by an optimization procedure for 4:he one-dimensional unit. In so

doing, two relations were found to be useful. They were a ime constant,

* - a SK' (3. i-1a)

and a ('ow-rate dependence,

Aq - /a (3.1-lOb)

both of which were inferred from approximate analytic expressions. They show

that the only two values, t 10 d and Aq 15 ft 3/d, which may be inferred

from the experimental data are functions of the three hydraulic

parameters, a, S, and W'. Thus, the inferred parameter set (Table 3.1-1)

is, unavoidably, onunique.

Discussion of the Code Output. Two tables included in the input echo

(see microfiche) may be of interest to the reader since this is the first use

of radial coordinates in this document. One, entitled "Radial e-id Block

Data," exhibits nodal points and grid-block boundaries. In this case these

radii have been internally generated (RI * in Card R-22) assuming an equal

ratio:

aril /r (3. 1-Ua)

for neighboring nodal points. That ratio is

An/2 r /r or A = 1.16597 (3.1-1lb)

where n X 50 (Card M-3-1), re ' RE 2000 ft (Card R-22) and r - RI -

1.0 ft (Card R-22). That all neighboring radii have this same constant rati

rty be verified by the reader.

The other table is exhibited under the title "Data for Carter-Tracy Wate

Influx Calculations". It shows the so-called terminal-race-influence functic

P I (Van Everdingan and Hurst, 19491 as a uncti:n of time. Implicit in this

101 I I I - -I

10'

a A#AbA 6/_ FLOW RATE

- X --- at *1 x w % s:1 0

. .-.4 6 in a -~~~~~~~~~~~e0

-1600

I.-,1M

o:

BL

10'!60 WATER LEVEL

LUWATER LEVE~L 6

- 500

10' -S. a

-Iw

-J

t1- tobIco0

10"-^ SWIFT* OBS, DATA aa

I 400

10'

I I I

104 10' l0' 10' 10

TIME (days)

Figure 3.1-4. Flov Rate atid Draudown of che H2A SIug Test Within the Magenta

Dolomite Formation.

3-L9

function is the assumption that the interior aquifer under detailed numerical

study for r < re (RAQ) may be analytically continued to infinity by an aquifer

having the prescribed Kb (KH) and Ob (PHIH) parameters. The nfilence

function denotes, in dimensionless form, the pressure drop at t aundary

r re between the two regions which results from a unit fow across this

boundary. Carter and Tracy 19601 used this function to achieve an efficient,

though approximate, relation

eW *W b p (3.1-L2)

It expresses the boundary flux e in terms of the pressure-change p at r re

s .ich occurs during a given time step. Quantities aw (P) and b tPOO

according to this tchnique, are themselves functions fo the terminal rate-

influcent unctior nd, of course, are implicit functions of time. Other

relations for P may be used to correspond to, say, a finite external

aquifer. The function presented in the output table, however, which

corresponds to an infinite aquifer, i the only one which is available

internally within SWIFT. The Theory and Implementation Document (Section 5.2)

provides additional information on this boundary condition.

Perhaps the o:- crucial tables of all, however, are those entitled "Well

Operation Summary," which are printed at selected time steps and which provide

the desired results. For example, from the first such table,

-4 3 Q(t-10 d) a 1.48 x 10 Ib/d - 23.7 ft /d (3.1-13a)

and from the et-and

Q(tlO d) a 165 Ib/d - 2.64 ft /d (3.1-3b)

both of which are consistent with the computed results of Figure 3.1-X. The

detailed results shown in this figure were actually obtained by post-

processing a plot file (Table 12) ox. which the ell-summary data was written

automatically for each time step.

3-20

3.1.4 Exercises

Level 1. Using the Input-Daca Guide, the Theory and Implementation and


exercises:

Exercise I

Exercise 2

Exercise 3

rxercise 4

Exercise 5

From ables 3.1-1 and 31-2, what data are required for the

blanks labelled "1" through "8" in Figure 3.1-3? The data

called for here are the hydraulic properties of the global and

local units and may be verified by reference to the output

listing.

Explain the meaning of the control parameters identified on

Cards M-2, M-3-1 end M-3-2.

Explain the meaning of the welL-control parameters identified on

on Cards R2-6 and R2-7-1 and the physical parameter identified

on the second Card R2-7-2.

Examine the printer-control parameters identified on several R2-

13 Cards. Then check the output to see that they produce the

specified output tables at the times indicated on the

corresponding R2-12 Cards.

How would you obtain the well-summary table for intermediate

time values between 10-4 d and 10 d?



Exercise 6 Check the validity of Equatiors (3.1-10) for this application.

For example, ncrease the value of K' and run the simulation out

to about 10 I d. Are the flow rates increased and the duration

of the dual-porosity effect decreased in accordance with these

equat ions?

3-21

.

3-22

3-23

3.2 PROBLEH 5. ANALYSIS OF WELL-TEST DATA FOR THE MUSQUODOBQZT AREA,

NOVA SCOTIA (FINDER AND REDEHOEFT, 19681

3.2.1 Objective

e To show the feasibility of characterizing the delayed-yield effect it

pumping-test data from the Husquodoboit River Basin with a dual-

poLosity model and to illustrate the manner in which such a

simulation is carried out.

3.2.2 Descriprion of the Problem

Background. The setting for this problem is the usquodoboic River Basi

in Nova Scotia (Figure 3.2-1), and the aquifer o interest is the zone of

glacio-fluvial deposits shown in Figures 3.2-2 and 3.2-3. As a part of a

water-supply feasibility study for the city of usquodoboit Harbor, a pumping

test was performed. The configuration of the pumping well and the three

observation wells is shown in Figure 3.2-4, and the measured data are

presented i Figure 3.2-5.

Pinder and others (Finder and Bredehoeft, 1968, *f.d Finder and Frind,

19721, it. their numerical studies, used these data for calibration of a site

model, with which they predicted drawdowns over a 20-year time period (Figure

3.2-6). These investigations characterized the site by the two-dimensional

geometry and the transmissivities shown in Figure 3.2-5. A delayed yield,

which they observed in the measured data, was characterized by a temporally

varying storativity:

0.003 , t 0

S a (3x10 4)(10-t) + (6x10 3 )t, 0 ( t 10 min (3.2-1)

0.06 , t 10 min

Problem Statement. That this delayed yield ay aLso be characterized a

flow in a doubly porous media is the focus of this probLem. Here it is

assumed that the observed drawdowns derive from he distribution of fine sar

Oleto, 'A 0 * :: . Msusquodobolt River basin 1

*--- * *;8:sz35p>- Musquodobalt Harbour: *g 9 ~ Dartmouth

VC 0~~~~~~C

@ 50~~~~~~~~~s 0 so,

Scale in miles

Figure 3.2-1. Problem 5. Location Map of the Musquodoboit River Basin

(From Pinder and redehoeft 1968)).

3-25

Figure 3.2-2. Geologic Map of Musquodoboit Harbour Area, Nova Scotia

(Corrected From Pinder and redehoeft, 1966).

3-26

L E E DCENOZO C

PLEISTOCENE AND RECENT

El] RECENT ALLUVIUMW GLACIO-FLUVIAL DEPOSITS

PALEOZOIC AND PRECAMBRIANN

GRANITE AND SLATE 5

ScaleHorizontal: 1 in 948 ft.Vertical: 1 in a 130 ft.

AA

100 > M~~~~Fusquodoboit River a

>I~~~~~~~~~~~~~~~~ C@ _ltI'-5_ ) Vs s t94^ z;>6 >,^P^W~s b4L~ws~s > as ^ -

q 4LA '7it.4A.

;m.~~~~~~~~~~~~~Aj

-j jk y 4 AJ%~

Figure 32-3. Geologic Cross Section, tKsquodoboit Harbour Area, ova Scotia

* 491 Sol

SCtan @ESt

go I

3

Meiroi

S. lete2. I1gOIItl

a te~~IS 1, . t\

t t" e. T U.I fal \ls

I XI,* tel O.MU ft _ IO

v pum1l all ^ 0.913 Woas

WI ertt.* l, I t * It,. In - OeyIlnse wll .r * 300 ft.

ful~ter .11 a . X t*

Figure 3.2-4. Location of .- servation Wells and Characterization of

Hdraulic Propert ies.

10.

1.0

z

0

0.1 0

WE WEL

.01I0.1 1.0 10. 100. 1000. 10,000.

TIME (min)

Figure 3.2-5. Observed and Simulated ravdawas for ella 1, 2 and 3 Using a

Homogeneous Dual-Porosity Mode.

. . . . .

2A- -

DigitzI model rsAts(sefected tne stepsi

0

0

_el DO - - ,. _

LaLI

g

W., 0

OA4

. . .......

O1.101

.0 10.000 100,000

PUMPING PER100 mnutes)

1=1=00 1QOOQOOQ

Figure 3.2-6. Time-Oravdavn Curves Obtained From the Simulation of Pinder

and Bredehoeft 19681. I

3-30

and of coarse sands and gravels (denoted hereafter as gravels), that the value

0.06, determined by Pinder, represent the storativity of and the fine sands

and that the measured ransmissivi~ies represent the connected gravel lenses

within the aquifer. The delayed yield, hen, rather than coming from the

time-dependent storativity of Equation (3.2-1) is controLled by the

conductivity of the fine sands and, to a limited extent, the storativity of

the gravels.

tn order to demonstrate the delayed yield effects as simply as possible,

it is assumed that the drawdown cone is confined to Zone 1 (Figure 3.2-4) and,

correspondingly, that the time period of interest is restricted to times less

than 10,000 min, the time scale of the pumping test. For these limited space

and time regimes, an axisymmetric site model may be used.

Flow Equations. The equations depicting coupled flow in the coarse- and

fine-grained regimes are identical to those for coupled flow in the two

fractured regimes, which were employed in the previous problem (Equations

3.1-1 through 3.1-3). ere, however, the dual-porosity zone is present

throughout the system rather than confined to the immediate vicinity of the.

wellbore.

Input/Output Specifications. Data selected for this demonstration are

summarized in Table 32-1. As indicated in the discussion above, all of these

parameters are taken from the work of Pinder, with two exceptions. The

hydraulic conductivity of the ine sands and the storativity of the gravels

were obtained by calibration against the drawdown curve for Well Number

(Figure 3.2-5) for times t 10,000 min. Comparison of measured and

calculated drawdowns is the desiAd result.


Discussion of the Code Input. Although the two porosities here refer to

two different porous-media components, rather than the two different

fractured-media components of the last problem, the code inputs (Figures 3.1-1

and 3.2-7) are quite similar in some respects. Both problems use radial

coordinates with a Carter-Tracy aquifer-influence function (Cards R-27

through R-33) affixed to the outer boundary. There are differences,

however. The well produces in this cse (positive value of Q Card R2-5) itt

a rate control (IINDWI - in Card R2-7-l) rather than a pressure control.

3-31

T&IMe 3.2-1. ProbLem 5. Input Specifications.


Gravel s

Transissivity T 0.274 ft 2 /s

Storstivity S 1.95 x 10-5

Fine Sands

Conductivity K' 4.63 x a-7 ft/s

Storativity so 0.012 ft 1

Thickness a 2.5 ft

Location of Observation WelLs

Well I r1 100 ft

Well 2 r2 212 ft

Well 3 r 3 300 ft

3-32

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3-33 ORAvr

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3-35

Further, the nodal radii are specified here in Cards R-25 (control parameter

RI 0.0 in Card R-22) instead of being internally generated. Finally, local

one-dimensional units are here imbedded in each global grid block, as

indicated by the parameters 11B - 40 and IFD - 0 in Card ROD-3.

Results. Figure 3.2-d shows the end result of the calibration of

simulated to field data. For this purpose th- two curves labelled "Lower

bound" and "upper bound" were quite helpful. Both are single-porosity Theis

solutions which assume that the gravels provide all of the transmissivity of

Lhe aquifer. The lower bound uses the combined storativity of the fine sands

and the gravels, whereas the upper bound uses only the storativity of the

gravels. Thus, the system resonse initially coincides with that of the upper

bound but bridges over to the lower bound at a rate controlled by the

magnitude of the conductivity assumed for the fine sands. Figure 3.2-5 shows

calculated and observed drawdowns for ells 2 and 3, as well as for Well .

It would appear that in the vicinity of Well 2, the gravels are a bit more

dominant than for Welt 1, whereas, in the vicinity of WelL 3, the fine sands

are somewhat more dominant. Both Figure 3.2-5 and Figure 3.2-8 demonstrate

the feasibility of using dual porosity to chattcterize the delayed-yield

phenomenna .

Discussion of the Code Output. The latter also indicates that drawdowns

(pressures) within the fine-sands come to equilibrium with those in the

gravels after about 100 in of pumping. After that the fine- and coarse-

grained materials behave asha single-porosity media. This effect may be seen

in the output tables (see . -ofichte) which, in this case, are inappropriately

labelled as "Pressure Within tb:. Rock Matrix." Take, for example, the fine-

sand surrounding Well I (Btozk 5). At t - I mn (6.9 x 104 d) he pressure

drop across the block isAI.0 psi; at t 10 min (6.9 x LO- 3 d) the pressure

drop is 0.1 psi; and at t 100 min (6.9 x 10 4 d) this drop becomes negligible

at 0.01 psi. Because of the choice of an initial pressure p 0, the

pressures may be converted to drawdown using the relation

s - - p/0.433 psi/ft (3.2-2)

in order to spot-check the results shown in Figures 3.2-5 or 3.2-8.

10.

1.0 I v

z

0

0.1

LWER BOUND *FIELD DATA

-. SWIFT

0.1 1.0 10. 100. 1000. 10;000.TIME (min)

Figure 3.2-8. Simulation of the Observed Drawdowns for Well I Using a

Homogneous Dual-Porosity Model. The upper bound is a

singl*.-porosity simulation using the gravel storativity only.

The lower bound i a sing1-porosity simulation using the

r-torativity of both gravel and sand.

3-37

3.2.4 Exercise*

Level 1. Using the Input-Data Guide, the Theory and Implementation and

the microfiche isting of the output, as required, complete the following

exercises:

Exercise I

Exercise 2

From Table 3.2-1, what data are required for the blanks labelled

"I" through "10" An Figure 3.2-77 The data called for here are

the hydraulic properties of the gravel and fine sand. They may

be verified by reference to the output Listing.

In what blocks are the observation wells located? (int:

examine the R2-5 and R2-7 Cards). These wells correspond to

Wells 1, 2 and 3 and their function is simply that of

convenience. The appropriate grid-block pressures are placed

withii he "ell Operation Summary" so that they may be easily

tabulated and plotted. The reader is referred to the output

listing, where he may examine several of these summaries.



Exercise 3 Figure 3.2-8 shows the behavior of the drawdown to lie

intermediate between the two singLe-porosity curves labelled

"upper bound" and lower bound". f one increases the

conductivity of the fine sand, will it move the simulated

drawdown in the direction of the upper bound? Check your

prediction by rerunning the problem.

3-38

-

4-1

4 FLOW AND TRANSPORT THROUGH AN AQUIFEF WITH CONFINING lAYERS

4-2

4-3

4 1 PROBLEM 6. DRAWDOWN FROM A FULLY PENETRATING WELL IN A LEAKY AQUIFER

(HANTUSH, 19601

4.1.1 Objective

* To illustrate the use of the one-dimensional local units to simulate

the effects of a leaking aquitard.

4.l.2 Description of the Problem

Problem Statement. A well fully penetrates an infinit. aquifer and is

pumped at a constant rate. The aquifer is bounded from below by an

impermeable bed and from above by a confining bed or quitard. The latter

influences the aquifer to a moderate degree since it is weakly conductive and

contains some fluid storage. A schematic drawing is shown as Figure 4.1-1.

Flow Equations. There are two flow equations here, one for the aquifer

and one for the aquitard. The former is given by

T r(r a - brw - S -(4.I-1

and the latter, which assumes perp icular flow in the aquitard, by

K' ' s'i* (4.2-r)3 2 W st

Coupling arises through the flow at the interface between aquifer and

aquitard:

r a K' a -z 0) (4.1-3)

where A- 1/b is the interface area per unit of aquifer volume.

?.4.: N* '.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4

im **

Figure 4.1-1. Problem 6. Scheatic Diagram of a FuLty Penetrating

Constant-Discharge etL n a Leaky Aquifer.

4-5

Initial/Boundary Conditions. For the aquifer the initial drawdown is

zero:

*(r~t-O) * 0 r O0 (4.1-4s)

and the discharge at the origin Is:

lim (r 1) 2 r*O dr T

t >0 (4.1-4b)

For the aquitard the initial drawdown is also zero:

8'(r~z,t-0) 0

and the interface condit ion is

I r > , -b' z O (4. 1-5e)

slr,zwO,t) - r,t) r > O , t > (4. 1-5b)

where z is -astired positive downward from the auifer-quitard interface. At

the upper boundary of the aquitard, there is assumed to be leakage from an

overlying aquifer so that

s'(r,2t-W.0 -t ' r > O. t > 0 (4. 1-5c)

Input/Output Specifications. able 4.1-1 provides the input data, which

is taken directly from the revised Benchmark Problem 3.2 (Ward, 1984bl.

SWIFT, of course, uses a pressure-based formulation rather than the drawdown-

bdsed formulation assumed by the benchmark-problem specification, with the

transformation between the two given by Equations (3.1-6) and (3.1-7). Table

4.1-1 gives the values appropriate for both formulations. In addition, these

data are supplemented by a reasonable value of the wel radius. The output

specification, also taken from Benchmark Problem 3.2, calls for the drawdown-

versus-time profile at a radius of 117.4 from the withdrawal well for a timi

period extending out to t 106 s.

4-6

Table 4.1-1. Problem 6. Input Specificatior.i.


Aquifer Trmnsmissivity T 0.05 n2/sec

Aquifer Hydraulic Conductivity K 5.00 x 0 3 n/sec

Aquifer Thickness b 0 M

Porus ity 0.10203

Water Density 1000 kg/m3

Water Compressibility cw 0

Rock Compressibility CR 5.00 x 10 7 WI

Aquifer Sorativicy S 0.005

Aquitard Specific 3torativity St 0.0016 1S

WeLL Radius rw 0.1143 m

Aquitard Hydraulic Conductivity KI' 10 5 /sec

Aquitard Thickness L' 50 m

Aquitard Porosity 0' 0.3265

Well Pumping Rate Q 6.283 3/sec


Discussion of Code Input. Up to this point, one-dimensional local unit

conceptually have been placed interior to the glo~tl grid blocks in order to

simulate dual-porosity effects. Further, the units were assumed to provide

storage only with a no-flov condition assigned to one of its sides. Here, t

role of these units is expanded in that they are plaree5 external to the glob

aquifer block in order to imulate an aquitard. In addiion, they are given

zero drawdown (Equation 4.1-5c) at the external end i order to simulate

leakage from an overlying aquifer. figure 4.1-2 typifies the numerical

gridding by shoving both the global aquifer blo:k, and the one-dimensional

units of the aquitard, which are themselves subdivided into grid blocks.

Figure 4.1-3 contains the necessary code input for the ocal units. t

control parameter

IFD a -3 , ROD-3 (4.L-6a)

effects the external placement f local units for all 50 grid blocks. Cont

paifameter

KBC * l , RID-3-l (4.1-6b)

chess activates the external constant-pressure boundary condition, which is

specified as

PBD - O.C R1D-3-2 (4.1-6c)

Figure 4.1-3 &lso gives the outer radius of the simulated region:

r X RE - 2647 RI-22

The selection of this radius, just as for the two previous problems was a

matter of judgement. Hovever, the appropriate characteristic radius

r * (Tb'/K1)2 X 500 (4.1-7)

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4-13

eid provide sone guidance. This quantity, which is derived from dimensional

analysis, properly accounts for the fact that the extent of the drawdown cone

is controlled by leakage from the neighboring aquifer. It, therefore, differs

from the characteristic radius used for the two previous problems (Equation

3.l-9a).

Result. The numerical resuLts are displayed graphically in Figure

4.1-4. As shown, there are two analytic solutions, both of which are taken

from Hantush 19601. On2 is for small values of time t < (b')2S/ICG'

4x104s), and the other is for large values of time (t > 2(b') 2 S'( -

8x105s). The comparison between numerical and analytical results is

apparently quite reasonable.

Discussion of Code Output. Thia output illustrates the generation of

SWIFT printer plots*, a facility which provides a quick, inexpensive way to

display computed results. Since the code only plots data pertaining to wells,

observation wells are defined in Cards R2-4 through R2-7. Further, since only

one plot per well is possible, three such wells are completed in BLock 35 at

the specified adius r 117.4 m. The resulting plots then display the timea

periods

0 t 2xlO s 0 C t 4x10i & and 0 t C 10 (4.1-8)

with different scales of resolution, which permits a detailed comparison with

the Hantush analytic solutions.

Plotting is activated by a positive value of the plotting key KPLP - ,

(Card M-2), whose value appears in the initial table of the output listing

(see microfiche). Note, in the three plots, which appear at the end of the

execution, that, due to the expanded scales of the ordinate, the difference

between simulated () and analytic () is readily appar3nt. Exercise 4 deals

with this relatively small (less than 0.4 percent al 106s) discrepancy.

* The term "plot" should not be confused with :he term map". Maps onsist

of two-eimensional contours of pressure, isopleths of concentratic.. and

isotherms of temperature. In contrast to plots, the mapping facility is,

in no way, coupled to the resence of wells.

ol- 0, LARGE TIME

E-X SOLUTION

z

0

10S SWIFT

0- ANALYTICALSOLUTION

S

10 10'1O 10' 101 10°TIME (SEC)

Figure 4.1-4. Graphical Comparison of the Numerical Results from SWIFT and

the Analytical Solutions.of Hantush for a Radial Distance

of 117.4 no

4-15

4.1.4 Exercises

Level 1. Using he Input-Data Guide, the Theory and Implementation and


exercises.

Exercise 1

Exercise 2

Exercise 3


"I" through "10" in Figure 4.1-3?

Consider Card ROD-3. ow would one simulate two confining beds

of the same thickness? How would one include differing

thicknesses?

Refer in the microfiche to the table f pressures within the

aquitard for radius r - 117.4 m, Block 35, and time c - 04 o

Convert to drawdown via s (p-p )/P(g/lg) ane plot. The

initial pressure, as given by PINIT in the table entitled

"Initial Conditions" is po 5x105 Pa*. Nodal positions are

given in the table "Specific Matrix Discretization Parameters"

located in the input echo. Has the aquitard reached steady

state? Repeat the plot for time t 106 s. Has the aquitard

now reached steady state?


code should consider the following exer'ise.

Exer. se 4 In the aquitard the local mesh was generated using single-mesh

generation (KGRD 1, RID-2), but was that the best option

available? In the local mesh, Node 20 is attached to aquifer

* Since SWIFT plotting routines do not permit negative pressures, the

initial pressure p which her i rbitrary. was set sufficiently

high in Card R-16 (see also -ed R-3) that no negative pressures

would be encountered.

4-16

grid and at Node 1 a constant pressure boundary was

prescribed. From the output table "Specific Matrix

DiscretizatLon Parameters" (see microfiche), near the aquifer

the grid increment is 0.5 (input parni.-eter DSD, Card RD-2)

whereas between Nodes I and 2 the increment is 7.33 . The grid

seems quite lopsided in that so many nodes are situated closest

to the aquifer. The reader shouLd try to improve upon this grid

either by using double-mesh generation or by specifying the

individual increments directly. One would not expect any

difference in the short-term response. The long-term response,

however, may be affected, but to what degree?

4-17

4.2 PROBLEM 7. HEAT TRANSPORT DURING FLUID INJECTION AVDONIN, 9641

4.2.1 Objective

O To illustrate the convective heat transport hich results from

injection into an aquifer with heat losses to the confiing beds.


Problem Statement. An incompressible fluid of temperature, T, is

injected into a confined aquifer of temperature, To, t-ough a fully

penetrating well (Figure 4.2-1). Both thermal convection and thermal

conduction occur within the aquifer, and thermal conduction is operative

within the confining layers, denoted herein as the over/underburden.

Transport Equations. For the aquifer, the transport equation is

-vpcp r +, r ar (r -) r pc - (4.2-1)

and, for the over/underburden, the transport equation is

Ks 32L + r P9_ 3't (4.2-2)M aZ2 H P

Here the over/underburden coupling term is given by

r a -2/b)K 3T (4.2-3)

and the fluid velocity by

v a Q2wrbP0 (4.2-4)

Initial/Boundary Conditions. The conditicns imposed on the solutions of

these transport equations are, for the anuifer,

T(rt0O) T , r ) 0 (4.2-Sa)

T(rO,t) TI t > t4.2-Sb)

* 1a~~~r...

* :..

- ~ ~ ~ .~~~ WER COFINING BE8K' PR P PR

Figure 4.2I1. Problem 7. Rdist eat Transeort Within an Aukfer with Losses

to the over/Udx L 1dn

4-19

and, for the over/underburden,

T Mxxt-0) T , z 0 (4.2-6a)

T'tX.-t-,t) * T t i. (4.2-6b)

T'(x,*O,t) T(x) t > 0 (4.2-6c)

and

T'(xz-bt) - . , t 0 (4.2-6d)

where is measured positive downward from the lover aquifer boundary.

Input/Output Specifications. Table 4.2-1 displays input specifications

taken directLy from the enchmarking Problem 5.1 [Ross et al, 19821. It also

contains supplementary data necessary for SWIFT input. Aside from the well

radius, for which a reasonable value is choser, the composite heat capacity i

partitioned as follows:

PmpMPC pc + O(1-)pRcpR/ (4.2-7)

Given the composite values (subscript, ) in the benchmark specifications, ti

values for the rock density and the rock heat capacities were chosen to be

consistent with this relation. The output specifications, also taken from

Benchmark Problem 5.1, calls for the thermal breakthrough at a radius of 37.

m from the injection well for a time period extending out to t 1O s.


Discussion of Code Input. Similar considerations arise both within th

aquifer and within the over/underburden. For computer efficiency, it is

desirable to minimize the outer extent of the simulated system but, yet, at

the same time, to adequately minic the infinite physical system. In additi

and for the same reason, it is desirable use as coarse a mesh as possibl

in the spatial and temporal regimes and stll to adequ..zly characterize th

thermal behavior at a radius r - 37.5 m frow the injection well. Within h

aquifer the controlling mechanism is the flow since i provides the convect

for the heat transport. Setting up the proper fluid velocity is no probleff

4-20



Injection Rate

Injection Temperature

Initial Temp.rature

Over/Urderburden

Thermal Conductivity

Density

Heat Capacity, Composite

Poros ity

Aqui fer

Thermal Conduct ivity

Density

Heat Capacity, Composite

Thickness

Porosity

Heat Capacity, ater

Heat Capacity, Rock

Well Radius

QTI

T0

10 kg/s

160 C

170 *C

K'

P'

PMC'

20 /(M C)

2500 kg/in3

1000 J/(kg 'C)

0.2

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PM

cpM

b

4cp

CR

rw

20 W/(m CN

2500 kg/m3

1000 J/(kg *C)

100

0.2

4185 J/(kg 'C)

2.079 x 106 J/(m3 C)

0.766

4-21

since, according to Equation 4.2-4), the flow is at steady state. This is

done by iposing a constant pressure at the external radius ., Card Rl-26-2 of

Figure 4.2-2 (see boxed value).

This external radius, of cou:se, ust be sufficiently large to contain

the movement of the thermal front which results from this flow field. Here

the retarded velocity is

v vocp/Dcal a v /r (4.2-8a)

-he., from Equations (4.2-4) and Table 4.2-1,

v Qc 2rbu c 1.332 x 104 /s (4.2-8b)o p iuPM

and the convective movement of the thermal front in t l09 s is

r * (v t)/'2 370 (4.2-9a)

To allow for a conductive smearing of this front and to include a slight

safety factor, the external radius is chosen to be

r * 1000 m , RI-22 (4.2-9b)

Gridding criteria are unknown in this case, and consequently it is necessary

to use a trial-and-error procedure for the spatial, and temporal, regimes.

Within the over/underburden the only transport mechanism of significance

is thermal conduction. The characteristic length corresponding to the time t

9 is

- (K-t/p c/. 90 n (4.2-lOa)

and the value

a a 300 m , RID-2-2 (4.2-lob)

is adorte; for the simulated thickness of the over/underburden. Here the

sp3tiol gciiding is chosen to adequately resolve the expected behavior within

4-22

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4-24

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4-25

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4-26

the over/underburden. This behavior, based on comparable solutions for a one-

dimensional system, should look much like a complementary error function which

is centered at the aquifer interface and which becomes ore and more diffuse

as time increases.

Results. Figure 4.2-3 displays the numerical solution alongside the

analytical solution of Avdonin 19641. Comparison with the case of heat

transport only within the aquifer indicater that the effect of conduction

within the over/underburden is relatively small. Consequently, the slight

discrepancy between numerical and analytical results most likely would be

attributed to the presence of some numerical dispersion, which arises in the

simulation of convection within the aquifer. Exercises 5 and 6 focus on this

point.

Discussion of Code Outpuc. The microfiche provides a comprehensive

listing of the code output. It includes the following: (1) an echo of the

input data, (2) preliminary setup computations for the radial grid within the

aquifer and the linear grid within the over/underburden, (3) time-step and

well sumaries, (4) a steady-state pressure distribution and (5) time-

dependent temperature distributions within both aquifer and

over/underburden. The reader should carefully examine this output, noting

particularly those data which are pertinent to Cuss, a heat-transport problem.

4.2.4 Eercises

Level 1. Using the Input-Data Guide, the Theory and mplementation and

the microfiche listing of the output, a required, complete the following

exercises:

Exercise I From Table 4.2-1, what physical parameters are required for the

blanks labelLed "1" through "3" in Figure 4.2-2? Check the

answers by referring to the echo of the input data printed at

the beginning of the output.

Exercise 2 Explain the function of the control parameters enclosed in boxes

in Cards -2, M-3-2 and R2-2.

Radius 37.5 meter! . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.

, ~. . . . . . . . . . . . ....... .... ...... . .. , . . .

0 5 10 15 20 25 30 35 40

TI ME (year)

Figure 4.2-3. Temperature Breakthrough Within the Aquifer at 37.5 from

the tnjection Welil.

4-28

Exercise 3

Exercise 4

Hov are steady-state velocities achieved as required by the

analytical solution? The solution option NCALL -2 (Card H4-2)

results in a coupled transient pressure and temperature

solution. owever, a steady-state pressure solution is achieved

by assigning a very small value of water compressibility. Could

a tero value have been entered? Before the reader attempts

this, he should remember that a non-zero accumulation is

required by the transient solution option. Would it be more

accurate to first solve a steady-state isothermal velocity field

(NCALL 4) and then switch to a coupled transient simulation

(NCALL -2, Card R2-l1.5)? If the value of water

compressibility is small enough .here should be no significant

difference.

Interpret the plotting data shown on Cards P-2 through P-4

(Figure 4.2-2). Also note the value of the plotting control in

Card M-2.


code should consider the following exercise:

Exercise 5 The source of the deviation of the numerical curve from the

analytical curve in Figure 4.2-3 is the issue here. As shown in

the R2-12 Cards of Figure 4.2-2, after an initial time step

of At 104 s, automatic time stepping is used, based on a

temperature increment of AT 1.0 C and a relatively large

minimum time step. Could the coarse time step selected by the

code be the source of the observed discrepancy? The reader is

encouraged to reduce both of these controls, rerun the problem

out to. say, t a 2x108 and then examine the effect of these

changes.

4-29

Exercise 6 Also, the temperature distribution within the over/unierburden

may not be resolved adequately at the time t 2 8 s, where

the discrepancy first begins to appear. From the output tables

labelled "Temperature Within the Rock Matric:", examine this

distribution at r - 37.5 (Block 1) to see. if there is a need

for additional resolution at this partiutar time. Reruo the

problem if necessary.

4-30

5-1

S FLOW WITK A FREE-W.TER SURFACE

5-2

5-3

5.1 PROBLEM 8. THE DUPUIT-FORCHHIEIMER STEADY-STATE PROBLEM EAR, 19721

5.1.1 Ob ective

* To demonstrate the simulation of steady-state flow, in a phreatic

aquifer.


Problem Statement. The following idealized problem is considered here:

A phreatic aquifer of length L (Figure 5.1-1) with fixed free-water elevations

h(x-O) ah (5.1-la)

and

h (xuL) * h L (.-

at its boundaries is subjected to surface recharge at a rate q. The problem.

is to determine the elevation h(x) of the free-water surface at positions

interior to the boundaries subject to the data given in Table 5.1-1.

Analytical Solution. To solve this problem analytically, the Dupuit

assumption is invoked (Bear, 19721. This amounts to a neglect of vertical

flow:

u a -K H/az - 0 (5.1-2a)

and yields a strictly one-dimensional solution in the x coordinate. Here is

the total head, H h - . Further, it is assumed that for the horizontal

flow, the ransmissivity is controlled by the saturated thickness:

T aK (5.K-hb)

so that the total discharge through a vertical surface per unit of width in

the direction (Figure 5.1-1), is given by

u -T Ia (5.13)

RECHARGE

%AI

.D.

ho

Figure 5.1-1. Problem 8. Schematic Diagram for the Dupuit-Forchheimer

Problem. I

5-5



Surface Recharge q 7.505xlO0 /s

Lateral Conductivity K 0.03 m/s

Vertical Conductivity Kz 0.003 /s

Height of Free-Water Surface at x 0 ho 0.75 m

Height of Free-Water Surface at x L hL 0.25 m

Length of System L 20

5-6

The equation of continuity Bear, 1972] then becomes, for steady state,

a (}C to ah) + q 0 (5.1-4)

rhich is the Forchheiter equation. For the boundary conditions prescribed in

Equation (5.1-l), the analytical solution of Equation (5.1-4) is the Dupuit-

Forchheimer parabola:

h2 h (h - h2 )x/L + (q/K )(L-)x (5.1-5)a L a

Flow rates at the two boundaries are obtained from Equations (5.1-2b) and

(5.1-3)

(0) (KI2L)h - h1) + (<))(qL/2) (5.1-6)

Here, the first term (on the right-hand side) represents the flow caused

strictly by the difference in heads on the two ends of the system. This rate,

of course is the same on both ends of the system. The second term arises from

the surface recharge. It shows that half of this recharge exits through the

boundary x 0 d the other )If through the boundary x L.

5.1.3 Numerical Solution

To solve this problem numerically with SWIFT, a two-dimensional vertical

cross section is gridded as listed in Cards R2-17 through R2-19 (Figure

5.1-2). Hydrostatic conditions

p - P - (gtg )h - constant (5.1-7)

are then prescribed for both side boundaries (Cards R-27 and R1-28). Hero

the vertical discretization is necessary for two reasons: Firstly, it

provides geometrical resolution for the free-water surface. Secondly, the

vertical discretization is required since the SWIFT formulation does permit

vertical flow. Thus, there is some difference at this point between the one-

5-7 /

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5-10

dimensional analytical treatment and the two-dimensional numerical

treatment. In general, the effects of the vertical flow may be minimized by

making the vertical conductivity, a free parameter in this case, sufficiently

large. Fot this problem, however, such was not necessary, and it as

sufficient simply to set K K

Insofar as the horizontal transmisivity is concerned, however, the

numerical reatment is very close to the analytical treatment. For a

completely saturated grid block, the full thickness A is used for the

cransmissivity. owever, for a partially saturated grid block, only the

saturated thickness h is used:

T Kah (5.1-B)x x

Results. Figure 5.1-3 shows the numerical solution plotted alongside the

analytical solution, and Figure 5.1-4 shows the same numerical solution

plotted in relation to the grid blocks. An evaluation of the flow yields

-3 3UL (SWIFT) a 1.10 x 10 m Is/n (5.1-9a)

and

u (analytic) I.-' x 10 3 3s/m (5.1-9b)L

a difference of about three percent.

Discussion of the Code Output. The SWIFT results are taken basically

from the tables which appear near the end of the output listing (see

microfiche). Water-table elevations come from the table of grid-block

saturations, and the flow rate L comes from the table of aquifer-influx

rates. The former table is printed through control IPRT, and the latter

through control 106, both of which appear on Card R2-13. To interpret these

tables, however, requires rome additional data from he output. For example,

by examining the former table, we note that, among the unsaturated blocks,

(6,K*5) w 0.8943 (5. 1-10a)

1.0

.8 - ANALYTICAL0 SWIFT

.6

.2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.

0 2 4 6 8 10 12 14 16 18 20X (m)

Figure 5.1-3. Steady-State Free-Water Surface far the Dupuit-Forchheimer

Probl em.

RECHARGE

IXI I .X I I, I I I

z

F

a

A

Figure 5.1-4. Ceometrical Characterization of the ridding Boundary

Conditions (Labelled "AIF") and the RecharRe. The numerical

solution is also shown in relation o the ridding.

5-13

is the aximum saturation. Grid-block tables, located near the front of the

output, are then used to convert the grid-block saturation to the elevation

h (x 6f5.5 ) 0.795 (5. -lOb)

To interpret the aquifer-influx rates requires the use of another table

of the input echo, namely he one for aquifer-influence block numbers. This

table shows that for the boundary x - 0, the aquifer-infLux blocks are

labelled "1" through "15", whereas, for the boundary x L, the boundary of

interest here, they are labelled "16" through "20". Consequently, to obtain

the flow quoted above in Equation (5.1-9), we form the sum

20 3 3UL Nm 16uK 1.10 k/s/mn a 1.10 x 10- M s/rn (5.1-il)

5.1.4 Exercises

Level . Using the Input-Data Guide, the Theory and Implementation and


exercises:

Exercise I

Exercise

Exercise 3


"1" through "4" in Figure 5.1-2, the input data set? Check the

answers by referring to the echo of the input data printed in

the output.

Interpret the boxed control parameters of Cards M-2, -3-t, 2-

1, R2-2.5 and 2-1l.

Interpret -he control parameters VAB and then analytically

verify the boundary pressures P, as indicated in the R-28

Cards. Use Equation (5.1-7). Also verify that these data were

read correctly by the code by examining the tables entitled

"Constant ressure, Temperature, Concentrations Block Types" nd

"Constant Boundary Pressures".

5-14


code should consider the following exercises:

Exercise 4

Exercise 5

Exercise 6

In the problem specifications the flow is essentially

horizontal. To what extent can the simulation be modified

before the Dupuit assumptions are in error? The problem as

specified has a maximum free-water surface slope of

approximately 10. Bear 1979, p.78] indicates that the error

is small as long as 2 (< I where is the slope of the free-

water surface. The reader should challenge the code by

increasing t slope and examining the error. This could be

accomplished by increasing the recharge rate, decreasing the

horizontal domain, and/or increasing vertical domain. An

increase in the vertical conductivity may be required for some

of these tests.

Can the free-water surface option be used in cylindrical

coordinates? CertainLy, in fact, Bear (1972, p. 311 presents

the analytical solution with Aich to compare. The reader is

encouraged to switch to a cylindrical g d (TG 3, X-3) and

execute this problem. We suggest a no-flow condition at the

perimeter of the aquifer and a pumping well at the origin,

completed in multiple layers, and subject to a specified bottom-

hole pressure of zero.

Can the free-water surface option be used for stratified

aquifers? Yes. The reader is encouraged to design such a

simulation in a Cartesian geometry (HTG 2 in Card -3) for

comparison with the discharge formulas of Bear 1972, p. 370

ff1. The reader should introduce heterogeneity by adding

layering in the RI-2t Cards.

5-15

5.2 PROBLEM 9. THE BOUSSINESQ TRANStENT-STATE PROBLEM [EAR, 19721

5.2.1 Objective

0 To dmonstrate the simulation of transient-state flow in a phreatic

aquifer.


Problem Statement. The following idealized problem is considered here: A

phreatic aquifer (Figure 5.2-1), semi-infinite in length, is initially

saturated to its full thickness so chat

h(x,tsO) h x > 0 (5.2-la)

Immediately thereafter, however, water is discharged at a sufficient rate to

reduce the saturated thickness at one end to half its original amount, i.e.-,

h(x=Ot) - h2 , t > (5.2-1b)

There is no recharge through the upper surface. The problem is to determine

the elevation of the free-water surface h(x,t) as a function of position and

time for the parameters given in Table 5.2-1.

Analytical Solution. Here, just as n Section 5.1, the Dupuit assumption

(Equation S.1-2a) is invoked, and the horizontal transmissivity is taken to be

proportional to tne saturated thickness h. The equation of continuity [Bear,

19721 may then be written:

a (K h h) _a (5.2-2)

This is the Boussinesq equation. Polubarinova-Kochina [ear, 1972, p. 384J

has obtained a general solution for this nonlinear equation, which is

presented graphically in Figure 5.2-2.

INITIAL

h1= 1.G

IP"

Figure 5.2-1. Problem 9. Schematic T" .ram of the Boussinesq Problem.

5-17



Lateral Conductivity Kx 0.01 ma/3

Vertical Conductivity* K 100.0 m/9

Porosity 0.50

Initial Height of Free-Water Surface ho t.0 M

Height of Free-Water Surface at z 0 h 0/2 0.50

* Adjusted to a suitably large value to approximate the Dupuit Assumption.

2.0

hlh(o)

U'a

1.8SO

V 2h h(0)

Figure 5.2-2. Transienc-State Free-Water Surface for he Boussinesg Problem.

5-19

5.2.3 NumericaL Solution

Discussion of Code Input. For the conduction mechanism (left-hnd side

of EquaLion (5.2-2)), the discussion here follows that of the previous

problem, which may be summarized by the following three statements: Firstly,

a two-dimensional grid (Figure 5.2-3 and Card -3-1) is required for the

numerical simulation as opposed to the simple one-dimensional analytical

simulation. Secondly, the Dupuit assumption is not used by the numerical

formalism but may be approximated by choosing a relatively large vertical

conductivity Kz (Cord R-20). Thirdly, the assumption that the transmissivity

is proportional co the saturated thickness is used in the numerical model just

as in the analytical model.

For the accumulation mechanism, the nalytical and numerical treatments

are quite close. For the latter, the grid-block saturation is unity for each

saturated grid block, but, for the partially-saturated block, it is

i a WhIA (5.2-3a) -

where Ah measures the saturated thickness. By defining saturation in this

manner, it can be shown that he:-teneral expression of accumulation, as given

in the Theory and Implementation reduces, under appropriate conditions, to the

right-hand side of Equation (5.2-3a):

Gi (pS) Oa (5.2-3b)

Results. Figure 5.2-2 shows the numerical solution plotted alongside the

analytical solution. For the most part, the agreement is quite reasonable.

However, for the smaller values of time the numerical free-water surface does

tend to drop somewhat more rapidly then does the analytical solution. Most

likely, this deficiency could be corrected by a mare refined spatial mesh

andlor a larger value of the vertical conductivity.

Discussion of Code Output. As ndicated in the discussion for the last

problem, the height of the free-water surface is not a direct output of the

code, but must be clculated from the output. Consequently, it is of interest

5-20 0DRpi-

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5-20

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I I LIt 00000 0

O 0 0 a 0 0

D.0 0.0 1oo 0.O

I 1 111 00000 0

0 0 0 0 0 0

0.0 0.0 1000.0

I I tl1 00000 0

0 0 0 0 0 a

0.0 0.0 1000.0

I I III 0C000 0

I1

0.0

0

0.0

a

0.0

0

0.0

0

2-1) 4

RZ2-1 1

6.0 6.25E-041a

R2-13 1

RZ-1 R2-13 ;

I

4.0 6*25E-041

R2-1 t

4.0 62SE-04t

I

22-13 1

R2-tI t

t2-i

6.0 .2SE-041R2-13

R2-I I

a.o i.2SE-oi l

12-. 1

tZ-z ~I4.0 1.ZSE-0'..

12-1 3 ;

0.0

0

0.0

0

0.0

0

0.0

0

._____ __.___ ____._ __ __ __ - __- -__-_-_-_-_-_-_- -_- - -_-_-_-

Fivire 5.2-3 Continued.

5-24 a

L Z * S 5 6 7 a1Z1450 r01 356 If q iLZ 3 b 40123456 7s90123fstya0L234567eX0

* 0---- - - - - - -- - --a- - - - - - - -- - - - - -__

It01 1 0 0

Itoz I 31.e06

It a) I l I

I1o 0 01

o o

0.0

I I

o 1

o a

00

I I

o a

0 0

0.0

11L 0

0 0

0 0

OO0.0

0000 a

0.0

I12-I I

I2S.0 b.2SE-0 1

IItZ-12 I

RZ-1-SIOP I

11

a

a - - - * _ _ __--------_- a---- - _ *


5-25

to go through the appropriate procedure once again. This time, however,

rather than using the grid-block saturations, as was done for Problem , we

shall use the grid-block pressures directly.

Turn in the output to the input echo, and from Lhe output table entitled

"X-Direction Di tance to Grid-Block Center" (see microfiche), note that the

distance

x 0.025 (5.2-4a)

falls at the center of the Blocks I 3. This is one of the distances used

for Figure 52-2. Advancing to the pressure-at-elevation table for time

t - 9.77 x 10 a (5.2-4b)

for which

C - (a/2K x2 x 0.4 (5.2-4c)

we cbserve that

p(3,6) a -287.6 Pa (5.2-5a)

Thus the unsaturated block thickness is

hh(3,6) - 0.0293 (5.2-5b)

Since the block thicknesses are z 0.05 m throughout, we easily compute

h(x,t) - (15)(0.05) - 0.0293 - 0.721 (5.2-6a)

in agreement with Figure 5.2-4. Converting to

maximum elevation h 2 m, we obtain

a dimensionless scale with

h/h (t 0.4) 1.440

(5.2-6b)

in agreement with Figure 5.2-2.

SURFACE AT = .0976 sec.

Z 1 13 _ J 1 20

T _ _ __ ___

AIF--

o .025 .125 5.3

Figure 5.2-4. Geometrical Characterization of the Cridding and Boundary

Conditions (Labelled AIF") in Relation to the ??umerical

Solution.

5-27

5.2.4 Exercises

Level . Using the nput-Data Guide, the Theory and Implementation and


exercises:

Exercise I

Exercise 2

Exercise 3

From the description of the problem and Table 5.2-1, what data

are required for the blanks labelled "I" through 4" in Figure

5.2-3, the input data set. Check the answers by referring to

the echo of the input data in the output.

Explain the reason for the relatively large value of KZ (Card

RI-20) and verify the boxed numbers in Card R-28. Hydrostati,

equilibrium (Equation 5.1-7) is assumed here just as for Problt

8.

What is the total flow rate passing through the boundary block

Level 2. The reader who want. :o develop a deeper understanding of the

code shoulI also-onsider the following exercise:

Exercise 4 Convert the Problem 9 data set (Figure 5.2-3) to cylindrical

coordinates using a multiply-completed well with a bottom-hoLe

pressure of zero to establish the boundary condition. Rerun t

problem and plot the results using the same variables (hs'h and

& - (/2Kt) 2 r) and scaling a in Figure 5.2-2. Should the

radial results ie above or allov the Cartesian results? Why!

5-28

6-1

NOTATION*

ROMAN SYMBOLS

a thickness of prism or radius of sphere for the local subsystem

aW explicit portion of e, the boundary flow rate of the simulated aquifer

A radial grid block-ratio

AV global/local interface area specific to global volume

Am specific activity

b aquifer thickness

bo aquitard thickness

bW implicit portion of eV, the boundary flow rate of the simulated aquifer

cPR specific heat of the rock

CR compressibility of the pores

cw compressibility of the fluid

C concentratioa of radioactive (trace) components

* The terms "global" and "local" are used in this section. Typically the

global model is used to represent the fractures or the aquifer. It is

reginally connected and may be three-dimensional. Usually the local

submodel is used to represent either the porous rock matrix or the

confining beds. t is one-dimensional only.

6-2

d fracture half-thickness

D dispersion/diffusion

DM molecular diffusion in porous medium

D* molecular diffusion in water

elf boundary flow rate into the simulated aquifer arising from the aquifer

which surrounds it

g acceleration of gravity

9c units conversion factor equal to g for the English system and equal to

unity for the SI system

h head

H total head

I radionuclide inventory

kd radionuclide disCribution coefficient

IV dimensionless distribution coefficient, Kv Rkd

K retardation factor or hydraulic conductivity

L length

m density of radioactive waste, i.e., mass of radionuclide per volume of

w3ste

ns number of nodes within a local unit

p pressure where subscripts 0 nd I refer to boundary locations

6-3

Pt terminal-rate influence function. used for odel external aquifer

q mass source rate or fluid recharge rate

qw radion- :lide source due to waste Leaching

Q rate of fluid withdrawal from well

r radial coordinate

re external radius of aqifer model

r, extent of stress-relief fracturing

rw radius of wellbore

rl radius of skin or radius to center of first grid block

source term for release of nuclides from the waste matrix

s one-dimensional coordinate for the local units. In formal equations

s - 0 represents local/global interface. In code, however, a 0

represents the external boundary of the ocal subsystem.

drawdown

S storativity

Ss specific sracivity

S saturation

t time

C3 total leach time

6-4

tD dimensionless time

T temperature, transmissivity or leach duration time

u Darcy flux

U mass fow rate through radionuclide inventory

v interstitial velocity

WI well index

x,y,z Cartesian coordinates

GREEK SYMBOLS

aL longitudinal dispersivity

r global-to-local radionuclide transfer rate

rH global-to-locaL heat transfer race

rW globaL-to-local flow rate

6p incremental change in pressure over a time step At

Ah head increment

As spatial increment in s for the ' ' subsystem

at time increment

Ax spatial increment in xi where xi x x, x2 y and x3 z

A decay constant

6-5

dimensionless Boltzmann variable

fluid density

PR formation density

p volumecrit waste density, i.e., volume of waste per bulk volume

T radionuclide half-life

t t' ottuosity

0 porosity

SUBSCREPTS

m fluid-plus-rock composite material

r radioactive component. Subscript is suppressed whenever no confusion

should arise. If given as an integer, r is the component numbeL in a

chain of species.

w radionuclide source (repository)

x,y,z directional indicator

SUPERSCRIPTS

denotes local subsystem

indicates characteristic length

7-1

REFERENCES

Avdonin, N. A., 1964. Some Formulas for Calculating the Temperature Field of

a Stratum Subject to Thermal Injection, Neft' i Ga:, Vol. , No. 3, pp. 37-41.

Bear, J., 1972. Dynamics of Fluids in Porous Media, American Esevier

Publishing Co., New York.

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

Carter, R. D., and Tracy, C. W., 196C. An Improved Method for Caiculating

Water nflux, Trans. SPE of AIME, 219, pp. 415-417.

Dennehy, K. F. and Davis, P. A., 1981. Hydrologic Testing of Tight Zones in

Southeastern New Mexico, Groundwater, Vol. 19, No. 5, pp. 482-489.

Dillon, R. T., Lantz, R. B., and Pahwa, S. B., 1978. Risk Methodolog for

Geologic Disposal of Radioactive Waste: The Sandia Waste-Isolation Flow and

Transport (SWIFT) Model, NUREG/CR-0424 and SAND78-1267, Sandia Narional

Laboratories, Albuquerque, New exico.

Finley, N. C., and Reeves, H., 1981. SWIFT Self-Teaching Curriculum:

Illustrative Problems to Supplement the User's Manual for the Sandia Waste-

Isolation Flow and Transport Model (WIFT), NUREG/CR-1968 and SD81-04lO,

Sandia National Laboratories, Albuquerque, New Mexico.

Hantush, . S., 960. Modification of the Theory of Leaky Aquifers, J.

Ceophvs. Res., Vol. 65, pp. 3713-3725.

Huyakorn, P. S., 1983. FTRANS, A Two-Dimensional Code for Simulating Fluid

FLaw and Transport of Radioactive Nuclides in Fractured Rock far Repository

Performance Assessment, NWI-426, Battelle Memorial Institute, Office of

Nuclear Wste Isolatv., Columbus, Ohio.

7-2

INTERCOMP Resource Development and Engineering, Inc., 1976. evelopment of

Model for Calculating Disposal in Deep Saline Aquifers, Parts I and I,

USGSIWR-76-6L, PB-256°03, Natiunal Technical Information Service, Wash agton,

D. C.

INTRACO1N, l983. ;nternational ucLide Transpoct Code ntercomparison Study,

Sedis;'4 Nuclear ower Inspectorate, Stockholm, Sweden.

Mercer, J. W., and Orr, B. R., 1979. nterim Data Report on the Geohydroiofy

at the Proposed Waste Isolation Pilot Plant Site in Southeast New Mexico,

U. S. Geo. Survey, Water Resour. Inv. 79-98.

Pahva, S. B., and Baxley, P. T., 1980. Detection of Fractures from Well

Testing in Procetdings:_ Workshop on Numerical Modeling of Thermohydrological

Flow in Fractured Rock Masses, LBL-11566 and ONWI-240, Battelle Memorial

Institute, Office of Nuclear Waste Isolation, Columbus, Ohio.

Pinder, G. F., and Bredehoeft, J. D., 1968. Appication of the Digital

Computer for Aquifer Evaluation, Water Resour Res., Vol. 4 No. 5, pp. 069-

1093.

Pinder, G. F., and Frind, E. O., 1972. Application of Galerkin's Procedure to

Aquifer Analysis, Water Resour Res., VoL. 8, No. 1, pp. 108-120.

Powers, D. W., Lambert, S. J., Shaffer, S., Kill, L. R., and Weart, W. .,

1978. Geological Characterization Report, Waste Isolation Pilot Plant (WIPP)

Site, Southeast New Mexico, Vols. I and II, SAND78-1596, Sandia National

Laboratories, Albuquerque, New Mexico.

Reeves, M., and Cranwell, R. M., 1981. User's Mlanual for the Sandia Waste-

Isolation Flow and Transport odel (SWIFT) Release 4.81, UREG/CR-2324 and

SANDS1-2516, Sandia National Laboritories, Albuquerque, New Mexico.

7-3

Reeves, M., Johns, N. D., and Cranwell, R. ., 1984 a. Theory and

Implementation for SWIFT 11, The Sandia Waste-Isolation Flow and Transport

Model for Fractured Media. To be published as a SAND-NUREG report, Sandia

National Laboratories, Albuquerque, New Mexico.

Reeves, ., Johns, N. D., and Cranwell, R. ., 1984b. Input Data Guide for

SWIFT 11, The Sandia Waste-Isoltion Flow and Transport Model for Fractured

Media, to be published as a SAND-NUREC report. Sandia National Laboratories,

Albuquerque, New Mexico.

Ross, B., Mercer, J. W., Thomas, S. D., and Lester, B. H., 1982. Benchmark

Problems for Repository Siting Models, NUREG/CR-3097, U.S. Nuclear Regulatory

Commission, 138 pp.

SilLing, S. A., 1983. Final Technical Position on Documentation of Computer

Codes for High-Level Wste Magement, NREG/CR-0856, U.S. Nuclear Regulatory

Corm-ssion, 11 pp.

Tang, D. ., Frind, E. 0., and Sudicky, E. A., 1981. Contaminant Transport in

Fractured Pcous Media: . Analytical Solution for a Single Fracture, Water

Resour. Res., Vol. 17, No. 3, pp. 555-564.

Van Everdingen, A. F., and Hurst, W., 1949. Application of the Laplace

Transform-j:inn to Flow Problems in Reservoirs, Trans. SPE of AIME, Vol. 186,

pp. 305-324.

Ward, D. S., Reeves, ., and Duda, L. E., 1984a. Verification and Field

Comparison of the Sandia Waste-Isolation Flow and Transport Model (SWIFT),

NUREG/CR-3316 and SAND83-1154, Sandia National Laboratories, Albuquerque,

New Mexico.

Ward, D. S., Reeves, M., Huyakorn, P. H., Lester, B., Ross, B., and Vogt, .,

1984b. Benchmarking of Flow and Transport Codes for Licensing Assistance,

GeoTrans, Inc. and Teknekron, Inc., to be published as a UREG report.

7-4

8-1

APPENDIX

CONWERSION OF INPUT DATA FROM SWIFT O SWIFT I

-

8-2

&M.

IJAIA�,"Pr

1 z 1I 6 S r a

. - -__ __ _ _ _ _ _, __ _ _ _ - _ _ - - - _ _ _ _ _ _ _ _ _ _ _ _ __j~~~~f34S6?#tq a, Z 114jj; jftq~~~~~~~~~~~~~~~~~~~~~~~~~~~~~jjv~ ~ ~

t

Z I

I

S *htREPLKSVD NTOnUTO

717 9

I 0

I ,______

£0 'ICP4VD

11 PS

12 1 uli 116 J1A JItK

13 1 IMP

5 a'

Iii

N-I 1

NPLP N.T NFLCIUNIT LO, _-_

INCHIK' TIE ONT I EK4C AI

I"-t~~~~

Nl-I II

rmrl-. I

20-1 1Ito- I-I I

I10-1-2 I

It0-Z-I I

I

RO-Z-2 II

100-L~~~

e*O 1M0tC&TfS TAT THESE VAIALESARE SITUATED IN REVISCO FORMATF IELS.

R00-2 I

XlA ig 51 I90,- -t - 1 tR If -100-3 1

1

_jq to tI

N-I II

11-1 I

itl-Z 9

11-1.5 UII

N-" INOICATES 4 f Vl5%Lf FOR SwIFT.ll

R 1-

I

I

Rl -5

Z2 I RlJ I

RL-q Z3 t ~~~~~~~~~~~~~~~~~~41-8

I CtRTEC Of TO SPACE LCetRTONS CE!AI4 VARI&4LE NAMES HAVE SEEN 11UHC&TEDI 1…__ _ _ , _ _ , _ ._ _ _ _ - _ _ _ - - - _ _*_ __ _ _ , _ _ _ _ _ _ _ _ _

8-3

l 2 3 4 5 6 7 4~~~~~~~~~~~~~~~~~~~~~Xr56t~qll}*sstSnlz 49st? n1z% s49D oU3456M90tMASo 7040lt14 St.regltz4s,67ss0

* -- *--- - * - * - - - - -~~~~~~~~----- ---

t I

I Z I RIl-1 I

I t

? t I

I 1

2S I Rt-l I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I

30 * Rl-tS I I

11 I Rt-2,6 1

I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I-I

I t13 I Rt-S I

ii I Rt-tS r

1S) * RI-aGI .

36 I RL-I-I 1

34~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ t- ~1-2~ 1I .I

3d I Rl-22 tI S

'.0 * ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~R -Z S I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I

t I

z. t ~FPIt H4DO rHADORl-z6-Z II ,___ ____,**** **.,.:9*,,*** I

'1 I fTUE Flu? FTUZ Rl-26-3 I

'.4 I * 1-27 ...... 1

Rl-ZS-I ...... *I N....

'b I Rl-20-2 .....1

'.t I 11-29 ..... I ......

'.6~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1-30-i I...... I .....

50 * ftI-31 .....I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

83-4

t 84 5 &I e

I z 3 1 S tzl 7sNq IMU'P. V04o0 31450?iq'z)*s674oI 9 2s6 2345 6 Oz ?A OLZ#shSqoI zr s607aqo

,_________,_____.___,_________._________._________,_________._________,__________.

i I tI51 I ~I-3t I

2 I S Rl-33-1 If I

S) I IRKCKSAYBK ID-3-1 II -I-# -0* -0*- - -

59 1 CR0 IO-I 1

S * -- ---- *--- - -_ -----.. ________..___

21 -I

5b t KCD sCRD SAD O5o OS0o RlD-Z-2 1

5? K OS 1Q-2 1

SE I IR KIIC KPS 'IPS KSIse 4lO-3)- 1 1

Ss I P80 rpeo sSuO 110-3-2 1

60* OIS9 ftL0-4

et I

625 1ZA-I o * I K I

i I

70 RIA-b

I t

1 RIA-02 I

ObI KIt 6? I ftl-3 I

61 I .hA-9 I

eq i COetH RLA-5 II ._______* .K

70 * ISA-b

I I7S I CLA-C K

I JJ2 K I HL R-OS t

I.--.~~~~~~~~~~~~~~~~~~~~~~Rl- It I

75 . 1K5c JOsc COIc iF-Il.___ ___ ___. l~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

._ _ __ _ ._ __ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ . _ __ _ _

8-5 Pr

l 2 3 1 5 h a

12 )-sfi7490tl4sbA9otz )*s7490z34sr6 ?6otzj4AhreQ~tz ag0~t}s~sl24hto#---------*--------- -______._______ .- - - - - _ - - - - -

71

7

la

III

I

t II t JI JZ £CHOI ._ _. _ .___. _______.1

t

INC" RI-I II

R -Z

60 '

81I

e8 I

Ia I

6%I

66 1I

U? £

qlI

ed II

69 II

qO

Iq IIQ2I

IRZ-Z.5 t

IAt-3 t

I

RZ-7-3 ;

I

I

I

R2- -L IRZ-1L II

RZ-7-1 KI

RZ-8 II

lRz-9 IK

Pi-to I

I

R2-LI.S II

kZ-L2 ~~I

ITWYPTKqTtXIYlCtYTlN IO TRUD.___ .__ +,_ _.. __. __--- _ ____.---

q

I

Sfp *NJRT~nRtt

b6 I XYXL XYTL XlXL %ZIL0 . _ _ _ _ _ *_ __ _ - - - ______ -- - _ _ _ _ .--------

q i II

Q I II

q I

(050 I080LItR~Z-93IOSO ~dDI ttRZ-1 3

If

K

It-t*ol II

11-IS II

P-.2 II

P-1 II I* -434fI N4U : t IEa OROAI XCG tKtR1. Z-11; 3 .E. C 'Cf 1 TO *IIWI I* _________*------- … .__-_____ . _______ .___ ___ - - _______ .__ __ __ - - - - - -- - -

8-6

MINIMAL CHANGES FOUIEO T CVERT a nTA SET FROM SWIFT I .4t ro

SWIfI I IZ. 3

CARD C14hC$s

P'-z JCD IF.fE 0 AND SHIfT VARIABLES PLC-IUNIt ro T tIcHtA0D LJloO

M-) SPLIT 0aJTo To CRDS, noOP A&&E, D0 NRCm-XuO. KmIEArO,KSLYOsO, NtDO. OItJoO

RI-it-Z ADO FP"IsO., Sfr HA0O &NO 1tuODO AIGCt

Rl-2h-1 ADD A LLNK FtU~aO. FTUVuD, FTUO1

ta-s nOU CnNVHsO.

R2-1 Aon 1RCH USE OfFAtLT8o

12-IL 50n IXVTP lZt, tXIC. tXYZC* 6UO, rUo (ALL SET a 0£

12-I) LCO 1050, 1050. I IIRTO (ALL St * Of

R-14 REVSED VARIABLES FOR EHANCED R&PPINC S NECESSARY

RZ-l§.S RkVISED VAMIABLES FOR EHRNCED APPING S ECESSARY

___________________________________.__________________________- - -

PROPOSED TAB TITLES

INIRODUCTtON

TRANSPrAT WITH DUAL

POROITY

PROB. 1: One Nuclide,

Frac./Prism. Matrix

PROB. 2: One uclide,

Frac./Spher. Hairix

PROS. : Nuclide Chain,

Frac./Prism. Matrix

FLOW WITH UAL POROSITY

PROB. 4: Well-Test in

Dolomui ce

PROB. : Well-Test in

Glacial Aquifer

FL1OW AD TRANS. WITH

AQUFERICONFINNG; BEDS

PROB. 6: FLow tIhru

Aquicards

PROB. 7: Heat Trans. from

Confining Beds

FLOW WITH FREE-

WATER SURFACE

PROB. 8: Steady State

PROD. 9: Transient State

NOTAT tON

REFERENCES

CONVERSION FROM

SWIFT TO SWIFT It

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