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Nagra Cédra CisraSocietà cooperativanazionaleper l'immagazzinamentodi scorie radioattive
NationaleGenossenschaftfür die Lagerungradioaktiver Abfälle
Société coopérativenationalepour l'entreposagede déchets radioactifs
TEGHNICALREPORT 8ÍI.1O
INTRACOIN LEVEL 1 BENCHMARKCALCULATIONS WITH EIR CODESCONZRA, RANCH AND RANCHN
J. HADERMANNF. RÖSEL JULY 1983
SWISS FEDERAL INSTITUTE FOR REACTOR RESEARCH,WÜRENLINGEN
Parkstrasse 23 54O1Baden/Schweiz TelephonO561205511
Nagra Cédra CisraSocietà cooperativanazionaleper l'immagazzinamentodi scorie radioattive
NationaleGenossenschaftfür die Lagerungradioaktiver Abfälle
Société coopérativenationalepour l'entreposagede déchets radioactifs
TEGHNICALREPORT 8ÍI.1O
INTRACOIN LEVEL 1 BENCHMARKCALCULATIONS WITH EIR CODESCONZRA, RANCH AND RANCHN
J. HADERMANNF. RÖSEL JULY 1983
SWISS FEDERAL INSTITUTE FOR REACTOR RESEARCH,WÜRENLINGEN
Parkstrasse 23 54O1Baden/Schweiz TelephonO561205511
NTB 83-IO 1
CONTENTS
FOREWORD
ABSTACT - Zusammenfassung
IntroductionLeve1 1 Benchmark Cases
Solution of the Transport EquationBoundary Condition EffectsDiscussion of ResultsReferencesTABLES
FTGURES
APENDIX
1
2
3
4
5
I2
3
4
7
13
T7
20
2L
23
40
53
-2-
vorwott
An der ÍNTRACOIN Studje nimmt das EIR als Projektteam tei7. Das Projekt-
sekretaríat wird vom Schwedischen Kernkraftínspektorat (SKI) ín Zusammen-
arbeit mit der NAGRA geste77t. Die technisch-wjssenschaftliche Atbeit
innerhaLb des Projektsekretariats wird von Kemakta Konsuft AB, Stockholm'
in Zusammenarbeit mÍt dem EIR durchgeführt.
Der vorTiegende Bericht wurde im Rahmen dev Zusarmnenarbeit EIR/NAGRA auf
dem Gebiete der Entsorgung radìoaktiver AbfäL7e eratbeÍtet. Er erscheìnt
gleichzeitig als NA3RA Techníscher Bericht NfB 83-10 und ElR-Bericht Nr.
491.
-3-
Abstract
We present the resuJts from calcufations of INTRACOIN Tevel- 7, case L and 2
(one-dimensional- advection-dispersion) benchmarks. fhe codes used ate CONZRA
and RANCH, corresponding to a semi-analgtical sofution of the transport equa-
tion, and RANCHN based on a fuffg numericaL soLutíon in the framewotk of the
pseudo-spectraL method. The inffuence of variaus boundarg conditions js in-
vestigated. Excellent agreement between resufts from the diffetent sol-ution
approaches is obtaìned.
Zusammenfassung
tntir präsentieren Resul,tate für INTRACOIN VergLeichsrechnungen der Stufe I,
Fäl-Le 7 und 2 (eindinensionaLer advektiver/dispersiver Transpott). Ðie be-
nutzten Rechenprograntrne sind CONZRA und RANCH, die auf einer semianaTgti-
schen Lösung der TtanspottgJeichung basieren, und RANCHN, das eínet volf
numerischen Lösung im Rahmen der pseudo-spekttaTen I,Iethode entspticht. Der
Einf|uss verschiedener Randbedingungen wÍrd untersucht. Die uebereinstimnung
der ResuJ.tate aus d.en verschiedenen Lösungsansätzen ist ausgezeichnet.
-4-
I Introduction
The importance of safe final- disposaT of radioactive wastes has l-ed to
extensive research and deveLopment activitjes jn mang countries. Fot
high-Jevel waste, repositories in deep geologic media are pJanned. InJong-term safetg assess¡nent the performance af the geologic barrierpTags an important roJ-e. In addition to the need of adequate data on the
characteristjcs of repositorg sìtes, appropriate mathematicaL model-s are
necessarv to describe the mechanisms invoJ-ved in the nuclide transporttram the repositorg to the biosphere.
To imptove the understandìng of important phenomena of radionucl-ide
transport and of various strategies for their modelTing an internatio-nal cooperation project /1/, /2/, INTRA11ÍN (International- NucLide Trans-
port Code Inteîcomparison), has been set up. The various mathematicaL
model-s of geospheric nucl-ide migration differ mainLg in two aspects. Theg
mag model different phenomena through díffering equations and theg mag
use various aLgorithms soLvìng these equations. Therefore, within the
INTRACOIN studg a comparison between far-field transport modefs is made
on thtee different Levels of increasing complexitg aimed at examining
l. the numericaL accuracg of the codes compared,
the capabj-Zitjes of model-s and codes to describe in-situmeasurements,
the quantitative impact of choosing either modeJling strategg on
the nucLide transport calcufations in a tgpicaL repositorg scena-
rio assessment.
At LeveL 7, calcuTations of various benchmark cases were performed with
some 20 computer codes (see tabLe 1). A finaL report on the resu-Zts js
in preparation bg the INTRACOIN project secretaríat (Swedish Nuclear
Poutet Inspectotate, Kemakta ConsuLtants Co, and Swjss FedetaT Institutefar Reactor Research) in cooperation with the participating parties and
project tearrs.
2
3
5
TABLE l- List of benchmark cases at Level 7 computed bg the vatious
project teams
Party Proj ec tteam
Cod e Benchmark Case No1234a4b56a6bT
AE CL
CEA/ T PSN
NAGRA
NRPB
PSE
SKBF/KBS
SKI
UKAEA/AERE
US DOE
VTT
hINR E
Ecole desMines
EIR
Polydyna-mic s
NR PB
TUB
KTH
Kem akt a
U KA EA/AE RE
Intera
0NV,l I
PNL
UCB
VTT
DRAMA XGARD2S XTRANSAT X
MET]S X
CONZRA/ XRANC H
RANCHN X
TROUGH X
GEOS X
ShIIFT X
TRUCHN X
NUC DÏF
COLUMN X
NAMlDNAM TA R
FT RANSSüIENT X
GETOUT X
MMT X
UCB-NE.X X
GETOUT X
MMT1D X
X
X
X
X X
X
X
x
X
X
X
X
XX
X
X
X
X
X X XXXa)
X
X
XXX
xXXXX
XXX XX
b) c) d)X X
X
a) No chain decag, sphetical tock matrix bTocks.
b) Axial dispersion omitted.
c) Dispersion omítted.
d) Chain decag and díspersion omitted.
-6'
In the present work we present Ëhe resuJ.ts of ealculatíons wíth theEIR codes R.AIVC¡I /3/, CONZRA /4/ and RANCHN /5/. In the next sectionwe gÍve, for the sake of compJeteness, the definití'on of benchmark
cases 7 and 2 at 7eve7 7. In section 3 a short description of thecamputer codes is presented. In sectíon 4 the ìnfluence of boundarg
conditìons js inr¡estigated. In section 5 we compare resuLts from thedÍfferent computet codes and dtaw some concJ,usions.
-7-
2 Level- L Benchmark Cases
As has been stated in the introduction, the LeveL L benchmatks have
been defined wíth the aim to compare code accutacg of different com-
puter codes using various soLution aLgorìthms. Cortesponding to
different modeL assumptions seven cases are distinguished:
one-dimensional- advection-dispersion with chaìn decag, constant
migration parameters (ground water veTocitg, retention factors
and dispersivitg) and constant leach rate.
one-dimensional- advection-dispersion in a Jagered medium
(piece-wise constant migration parameters) .
one-dimensionaJ- advection-dispersion with continuousLg varging
mígtation parameters.
Two-dimensional- advection-dispersion with constant retention
factors and dispersivitg.
a) Paralfef water veTocitg field and transverse dispersion.
b) Non-uniform water veTocitg fiel-d and transverse dispersion.
one-dimensionaf advection-dispersion with diffusion into the
rock matrix.
Two-dimensional- advection-dispersion with matrix diffusion.
a) ParalTef water veTocitg fiel-d and transverse dispersion.
b) Non-uniform water veTocitg fiel-d and transverse dispetsion.
One-dimensionaL advection-dispersion with l-ineat mass transfer
kinetics, chain decag and constant migration parameters.
Case l- incl-udes several parameter variations. Vle have performed
caLcuLations for case l- and 2.
I
2
3
4
5
6
7
-8-
2.1 Transport equation
For benchmark cases L and 2 the transport equation is
ð
arRx c ^/o
K:av#c* cK-L
(in particle units) of
, v Ís the interstitiaf
vd-'--C - I R C + ÀiJxKKKKK-f R
K-7(1)
(2)
where C--(x,t) denotes Èhe concentrationK
the K-th nucLide with decag constant X*
graundwater veLocitg, a the dispersivitg and R the retentionK
factor for the K-th nucl-ide.
The sol-ution to eq. (1) is subject to the initial- conditions
CK
(x,t=0)=0, K:l ]V
2.2 Boundarg conditions
The boundarg conditions, upstream and downstream, have to be
specified for soLution of eq. (7). Sone freedom exists in in-terpretation of the benchmark definitions, which has Led to
vatious impJementations of boundarg conditjons. The effect ofboundarg condition on resuLts will be dìscussed in the next
section.
2.2.1 Upstream
At the geosphere inlet, x = 0, two different boundarg conditions are
defined
a) Concentration boundarg (8l)
For each nucLide K a boundarg concentration is specifiedaccording to
K(t)
forveFT
It<T
cK
(0,t)0 for t>T
(3)
-9-
where ve? ís the water flow at the inlet and I*(t) ís the nuclide
specifíc ínventorg at time t calcufated from the Bateman equations
(ín particle unÍts)
# rK(t) = -\* ,*
¿2
ð;z x-l- c x-f
* \*-r'*-l (4)
( 7a)
(s)
Inventoties at t = 0 are gíven ín table 2. The leach tìme was takent
T = l-0' qr (I2).
b) Saurce boundarg (82)
The transport equation (7) ís nodified to incJ-ude a soutce term at
the geosphere ínLet
**#cK=av - x* **'* * \*-J *cK
d
ð"vK
c
FK(t)
EF6fx)
where F is a fTow tube cross section (bg definition F = 700 m2 ) ,
ôfx) the Dirac deLta function and F*(t) the ìnjection rate defined
bg F*(t) = IK(t)/T for t < T and zero eTsewhere. The introduction
of a source entaiTs the necessitg for definitíon af a boundarg con-
dÍtion. We have assumed that the medium extends to minus ìnfínitg*)and
C(--, t) = 0.
Hencet the phgsícal picture is that of a pTane souîce in a semi-
íntinite (x < 0) medíum.
*) We note that the assumption of a constant dìspersívìtg wìth
the same vaTue for x > 0 and x < 0 ís questíonabfe ín the
framework of a phgsical interpretation of dispersion.
-70-
2.2.2 Downstrean (87)
es ìt is the case upstreamt also downst.ream several vatiants ofboundarg condítions a¡e phgsicallg feasibLe (see e.g. ref. 5).
Ííe have chosen a semi-infínìte medium (x > 0) and
Lim-. C--(x.,t) = 0 .*d * - K' d
?his choice is dictated for the semi-analgtíc code RANCH. For the
numerÍcaL code RANCHN a suitabTe value for xO has to be chosen.
Its infLuence is discussed in the next section.
2.2.3 Interlager boundaries
Fot case 2 benchmarks interJager boundarg condítíons have to be
specified. These are
a) Cantinuítg of concentration
(6)
(7)
(8)
Jinô+o
7ìnô+0
C*(x.+ôrt)
âc-a:K )\dx
(x .-6,t)7ìnô+o c
K
whete x, denotes the cootdinate of the boundatg.L
b) Canservation of flux
{ev(cK x=x.+ô
J-
7inô+o x--x . -ôL
We poìnt out that eq. (8) is rìgorousLg fulfí7Led in the
RANCIIN calculations, i.e. no discretization dependent
smearing of parameters at the boundarg occuts.
In the RANCH mode7, eq. (8) is replaced bg
{ev(c*- " * ,,
" r,*(*rt) = 0 (e)
-17-
where the subscript ì denotes the soJ,ution of eq. (7) ín the
i-th 7ager. It can be shown /3/ that thjs is a good apptoxi-
mation to eq. (8) as Tong as Ui/ui >> 7 where [, is the mìgra-
tion distance ìn the i-th 7ager.
2.3 Parameter definitíons
For both benchmark cases twó independent decag chaÍns, denoted 17
and 12, respectiveTg, are defined. The data are given in table 2:
Nuclide qtlYr) IK (O ) [Act iv ity un its]
I1: long lived mother nuclide
234-U
230-Th
226-Ra
2.445 +5
7.7 +4
1.6 +3
1.000
0. 010
0. 004
I2z short lived mother nuclide
245-Cn
237 -Np
233-u
8.5 +3
2.14 +6
1.592 +5
0. 700
1 .000
0. 004
TABLE 2 NucTide data (half-LÍfes, inventoríes at zelo tìne)
-72-
Nucl-ide independent data are shown in tabLe 3:
Layer Migration Porosit,yl eng t,h¿ [mJ a
!'laterv e1 oc ityv lm/yr)
Di sper s iv itya Im]
Case 1 1 500
5000
(11 )
(12 )
0. 01
0.01
50
500
(P2 )
(P2)
Case 2 1 50
100
350
0. 01
0. 01
0. 01
1
1
1
5
10
35
2
3
TABLE 3 Geohgdraulic data for cases 7 and 2. The notatìon ofthe particuLat patameter vatiants is given in paran-
theses.
Two diffe,rent sets of tetention factars R have been defined:K
Set Layer Cm Np U Th Ra
Case 1 R'l 1
R2 1
5 000
60
700
200
300
60
20000
500
1 0000
20
Case 2 R1 1
R2 1
2
3
1 0000
5 000
2500
500
60
30
1 400
700
350
700
200
100
600
300
150
300
60
30
40000
20000
1 0000
5 000
500
250
20000
1 0000
5 000
2000
20
10
2
3
TABLE 4 Retention factors for cases 7 and 2. Set R7 nightcorrespond to reducing, set R2 to oxìdizing condi-
tions for case 7-
-13-
3 SoLution of the Transport Equation
3 .1 Analgtical- SoJ-ution
In generaT, analgtica] soLutions to eq. (7) ate not known. How-
ever for the first nucLide of a chain and a homogeneous medium
(case l-) a soJution can be given for the boundarg conditions eq.
(3) and (6) and the initiaf condition (2) (see e.g. ref. 7)
C (x,t)Ir_(0)IEVFT
{erfc ( f- ) + exp(x/a)erfc (f+ )
- s(t-T) ¡erfc(ø-) + exp(x/a)erfc(s+)I
whete etfc(z) is the complementarg ertor function and s(t-T)
the unit step function. Furthermore we have defined
(10)
( 11)
+¿- (x+ t', 4avt/R l
+s (x + ! r*rll Inf 4av(t-T) /n,
3.2 Semí-anal-gtic Solution
In the RANCH modeL /3/ eS. (1) is sofved for boundarg conditions(3) and (6), bg the aid of LapJace transforms. The sol-ution of
eq. (la) together with boundarg conditions (5) and (6), is per-
formed in two steps.' coNzRA provìdes the concentrations at the
geosphere inLet (x = 0) which are then used bg subsequent. RANCH
calcuLations. A short model and code chatacterizatìon is given
in the Appendix.
-t4-
The sol-utìon of the transport equations can be wtitten
K-1ci,g (*i't) {c (0,t-t)írK
t
I BKK'
KK,(0)irKB (t-r ) c (x.,r)dr
I
DK'=0
(t-t 1, "::,'- (x .,r )dr
K-1 t
Io
(12¡
(13)
(14)
+ tK,:O
where i denotes the Tager to be considered. The quantitrj c(,0)-- i"irKthe soTution /7/ for a puJ,se release and B**, nãg be regarded as
effective tepositotg concentratjons. Fot the present putpose itis important to note that the B**, are superpositions of terms
-l<9,tm
exP{-À (t-r)\ c -t tIIt(0,t)dr
t
Io
Kr,
whete
9.
R
RKK
L
R
R
KL
-t)L
If I\KL < 0, the ampTìtudes B*U Íncrease exponentiallg with tÍneand probJems mag arjse in RANCH calculations. This Ís especiallgthe case for tÍmes t l, ïO/LKU. Howevert this is actuaTTg not aserious dtawback. Fot actual cases, at those times, radioactìveequiTibrium between the probTem posing nuclide and jts ptecutsotis a good approxÍmatìon.
JLK
( À À )/(
lntegtation in eq. (I2) is performed bg the trapezoidaL rule. The
time steps are chosen sucå tåat the íntegration covers the fu77
tange where ,'r:'- , 70'20 with tgpicaTTg 2000 to 3000 integratíonpoinûs.
-75-
3 .3 NumericaL Sol,ution
Inthe Rå¡VC¡flV model /5/ eS. (7) is solved for boundarg condítions(3) and (6) using the spectral method. Herebg the solution is ex-
panded ìnto a finite series of Chebgshev poJgnominaTs TU(x) as
cK
(x,t) =M,
Ð,, - I ouo - ! au, t øf, r (x).
9,=0
(ts)
(16)
(17)
(18)
lnsertÍng ínto the transport equation and using the orthogonaLitg
condition for the Chebgshev polgnominal,s J,eads to the foTTowing
sgstem of ordinarg differentiaT equatíons
dt c*(9 U,t) =
Mtm=0
A c (gn,t) - \* "* (vu,t)d K
9,m K
**-j
K-I RK "*_j
(v u,t)
where the coupling matrìx is given bg
KLm î rr(v^)
2LR4av d2 d
Kdg
rn(v | +2vLR
Kdg2
L Ís the sgstem Tength for the solution of eq. (1) and the
coordìnate V, is defined bg
q = cos (nn/u)-m
and we have perfotmed a coordinate transformatÍon
+l
r,rvutl.A
M
Tn=0
x, = L/2 (7 - gn). (le)
-16-
3.2.1 Time Integration
Sjnce the differentiaT eq. (16) nag behave as a stiff sgstem we
use the backward Gear method /8/ for the numericaT sol-ution. LocaL
error estimates are possibl-e and we require a tefative accuracg of_Ã
l-0 - in our cal-cuLations.
Sjnce the time behaviour of the nucLides in the decag chain mag be
verg different, eq. (16) is not sol-ved as a coupJed set for al-L
nucLides simuLtaneouslg. Instead, the contribution of the precursor
(l-ast term in eq. (16)) is taken as a source term thus decoupTing
the different nucLides in the cal-cuLations.
3.2.2 Order of the Chebgshev Approximation
No rigorous mathematical- critetion an convetgence with respect tothe order ItI of the expansion exists. We have varied M in order toinvestigate the convergence. A tgpícaL exampJe for case 7, varia-tion ( 17, T2 , BJ-, EL, Lf , P2 , R2 ) , is shown in the foTTowing tabl-e
for the maximal- concentrations.
Order M
Max imal
234 -U
Concentrations
230-Th 226-Ra
of Max imum
230-Th 226^Ra
Time
234-U
10203040
.8177 0-5
.81502-5
. B1 505-5
. 81 505-5
.6262^7
.6225-7
.6225-7
.6225-7
. 21 17 5-5
.21 1 15-5
.21 1 15-5
.21 1 15-5
60000 130000 124ooo
TABLE 5 Stabilitg of maximal- concentrations with tespect ta the
order of Chebgshev exPansion
It tutns out that IuI = 40 teproduces at l-east four significant
digits for the case l- benchmarks and for the whole tíme inter-
vaf consid.ered. If one is interested in the concentration maxima
tegion, onLg, an otder of M = 15 to 25 ìs sufficient. For case 2benchmarks we have chosen M : 72t L0 and J6, respectivelg, forthe three Jagers.
-77-
4 Boundarg Condítion Effects
4.1 Downstteam
As afreadg ment.ioned we have chosen the EJ- boundarg condition, eq.. (6).
Fo;r our RANCHN caLculations this ìmplies ,*(*d,t) = 0 for a suitabfe
vaTue of xO to be chosen. We determined *d bg the requirement that the
maximal- concentration at the migration distance LI = 500 m ís indepen'
O is cJearTU
In order to investìgate the ìmpact of a particuTar vaLue for xO on
the time distribution it is appropriate to compare the RAIVCHJV resuJts
with the exact soJution, eq. (70). To this end we have seLected the
benchmark variation with the fastest runníng mother nuclide (R = 60).
The resufts for the 234-U concentrations at the different times are
shown in tabl-e 6, where t = 62000 g is the time of maximum.
dent from thìs choice. Sjnce tru* o R tåe minimal- vaTue of x
dependent on tetention factors.
xd (m)
234
t = 62000
U concentration
t = 200000 t 3 00000
9001 0001 1001 200
. B1 505-5
.81511-5
. 815 i 1-5
. 8151 1-5
. 486 1 3-B
.50717-8
.51261-8
. 5 1 450-B
.38253-1 2
.43934-12
.45038-12
.47 423-12
ex act .81511-5 .51471-8 . 48994 -1 2
TABLE 6 concentration of 234-U at dÍtferent tímes and fot vatlous
values of xU in comparjson to the exact tesuits. The pata-
meter variation is (If' f2' Bl-, 87, LL' P2, R2)-
It ìs seen that the maximum concenttation at time t : 62000 g is
sígnificant up to four digits wíth the choice xO = 900 m. FoT
Targer times devÍations to the exact resufts are seen, which de-
crease wÍth incteasing xO howevet. For the high retention factot
set (Rl) the value for xO could be chosen much smaLTer.
-18-
For the case L benchmarks we have fixed x = 900 m and for case 2d
benchmark xO = 800 m.
Let us consider first a comparison*) with exact resuJ,Ès fot the
first nucl-ide (tabie 7.7 to 7.3). We have defined
DEV(7.) = fOO* (RANCH(N) - EXACT) /EXACT
For the high retention factors (Rl-) smalT discrepancies buiTd up
sTowJg at Tatge times, onlg, and at Low concentration TeveLs. The
deviations are Targer for the l-ow retention factors (n2) as is ex-pected. At vetg low concentration Tevel-s the numericaL accuracg re-quirement is feLt. The verg same effect js seen when comparing
RANCIIN and RANCH resul.ts (figures 7.L to 1.4). Here, the infl-uence
of individuaL retention factors are exhibited: take as an exampLe
variation (I1,72, BL, Ef, L7, P2, RL) in figure I.7, where 234-U
has the Towest retention factor and sma77 deviations start buiTding
up at t \, 500000 gears. The other two nucTìdes, 230-Th and 226-Ra
have verg nuch higher retentíon factors and the resul,t vittuaLlgagree over the whol-e tíme span shown.
4.2 Upstream
The standard boundarg condition we have chosen is {, eg. (3).
For the Teach tine 705 g (r2) and the Peclet number 70 (P2)
chosen, differences to a source boundarg conditions, eqs. (1a)
and (5), ate expected to be smalL. It can be shown /4/, /9/ thatresul.ts from these two boundarg conditions differ most when theparameter 4aR-/v is Targest. ?hjs is the case for the retention-K
*) The resuLts of RAJVC¡íff calcuLations differ verg slightlg fronÈhose in ref. 72 sìnce here the relative accutacg of time
integration was chosen L0-5 and in the latter reference 70-4.
-79-
factor set R1. Consequentlg we have performed CONZRA/RANCH caLcuLations
for benchmark case 7, variation (17, T2' B2t El-' L7, P2, Rl-)' The maxi-
maf concentrations and the times of the 70% feveTs, '-.1
und T+.I' ""'pectivelg, are compared in tabJe I to the resu-Zts from a RANCH caLcula-
tion with BJ- boundatg condition-
Nuclide Boundarycond it ion
c T T +.7m'-. Lmax max
234-U
2 30-Th
226-Ra
B2B1
7. B0+46 .7 3+4
1 . B4+51 .65+5
5 .96+55.54+5
1 . 08+59.47+4
.297 1 -7
.3468-7
31 96-53935-5
.5941-7
.6936-7
2 .54+52 .28+5
2 . j4+52 .28+5
1 . 08+59. 47+4
3.7 4+53.27 +5
5.96+55.54+5
B2B1
a)B2B1
a) secuLar equiTibrium assumed
TABLE 8 Compatison of resu]Ès from diffetent upstream boundarg
conditíons.
The effect of boundatg cond.jtions is sma77. Fot 82, nucTídes arrive
7ater, the maximum is lowet (Less than 20%) and the time distribu-
tíon somewhat broadet. These characteristics are weIT undetstoad
since fot a pTane source boundatg concenttaÈjons at x = 0 vanísh
for time t = 0 and buiTd up with íncreasing tìme.
-20-
5 Discussion of Resu-Z,ts
we present the resuJ.ts of out calcul-atians for cases L and. 2 in formof graphs (cancentration versus time, figures I - 4) and. tabl-es, where
we give the maximum concenttation, the time of the maximum and thetimes T , and T. , where f0"2" of the maximum concenttations is reached.-.1 +. L
on the front and the traiTing part of the time distribution, respecti-veLg. DetaiLed tabLes of time dístributions are given in refs. /r0/ to/12/. In genetaL the agreement between RANCH and RANCHN cal-cul-ations jsexcell-ent.
Fot case 7 both caLculations differ bg less then 3%o in the vaLue ofmaximaL cancentrations. The shape of the peak region of concentrationcurves agrees sinì7arlg we77 (tabl-e 9). At. Targe times the particul_arÍmpJementation of downstream boundarg cond.ition is fel-t (see d.iscussjonin sectìon 4). Eor variation (If, 72, Bl_, El, Ll_, p2, R2), figure J.2,the RANCH caLcuLation of the fast-running short-lived 226-Ra posed.
probLems because of the reason mentioned in section 3.1-. rn this caL-cuLatíon the contributíon of 234-u to 226-Ra is neglected for timest > 150000 gears. From this a discontinuitg of 62% at t = l-50000 gears
resuJ, ts.
For case 2 benchmarks agreement between RANCH and RANCHN is somewhat
ress ptanounced. This comes from the approximation of interTagerbaundarg conditionst eÇ[. (9), in the RANCH ca]cul-ations. Because ofthe smaLler x. vaLue (sectíon 4.L) chosen in the RAN1HN cal-cuLationsddevíations at Target tímes are mote apparent. ALso here, the teTatíve7gTarge deviation for 226-Ra in variation (Il,72, Bl, Ef, R2), figure2.2, are stennning from neglection of 234-U contribution for times Targer
A
than l-.38 70- gears.
úíith RANCH we have performed additionaJ, case L cafcuLations for thelong migration path (L2). clearrg, the maxima are rowered. and thepeaks become broader (tabLe TL and figure 3). The ínfLuence of up.
stream boundarg condition, see arso fìgure 4, has been ìnvestigat:edand discussed in section 4.2.
-21
AcknowTedgement
We wouLd like to thank H.P. Aldel^, H. FJurg and Ch. McCombie fot theit
interest in this work, NAGRA for partiaf financial support and S ' Wittke
for a carefuf tgping of the manuscript.
Refetences
/1/ K. Andersson, B. Gtundfeft and J. Hadetmann, INTRACOIff - .An InteÎ-
national NucTide Transport Code Intercomparison Studg, in Scientí-
fic Basis for Radioactive Vlaste \Lanagement - V(W. Lutze' ed.),
Efsevier New York 7982, P. 839-
/2/ A.H. Larsson, K.A. Andetsson, B. GtundfeTt and J. Hadermann,
MathematicaT !ftod.eLs fot NucTide raansport in Geologic Media -
An International Intercomparison (INTRACOIN), Intetnatìonal
conference on Radioactive waste Management, seattTe, wA, USA'
16 - 20 I,Ias 7983, IAEA-1N-43/434 (1983) -
/3/ J. Hadermann and J. Patrg, IVucf. Technof. 54 (198I) 266 and
teferences quoted therein.
/4/ U. Schmocker, J. pattg and J. Hadermann, On the connection betuleen
injectíon rate and repositorg boundarg concentration in geospheríc
transport modefs:' in Environmental migration of Tong-7ived radío-
nuclides, Proceed.ings of a sgmposium, KnoxvilLe 1981, IAEA Vìenna
7982, p. 681.
/5/ E. Rösef and J. Hadermann, Numerical SoLution of One'Dimensional-
Transport ProbTems, EIR internaT report TIt'l-45-82-48 (7982)
/6/ J. Bear, Hgdrodgnamic Dispersion, in FLow Tzough Porous Media
(R.J.M. Wiest, ed.), Academic Press New Yotk 7969' p' 709, see
especiaTTg p. 786 ff.
-22-
/7/ J. Hadermann, Nucl. TechnoL. 47 (De}) 3I2
The Automatic Integration of Ordinarg Differential_Cornn. ACM 74 ( 1971 ) 744 .
/8/ c.W. Gear,
Equations,
/9/ LJ. Schmocker, J. Patrg and J. Hadermann, On the Connection
between rnjection Rate and ReposiÊorg Boundatg concentrationin Geospheric Transport ModeTs, EIR-Bericht Nr. 426 (JggJ).
/10/ J. Hadermann and F. RöseJ,, rNTRAcorN Benchmark carcurationswith code RANCH: Level- 7, EIR internal report, TM-45-92-3
(1/2/82).
/tl¡ A. Bundi, J. Hadermann, F. Röse7, EIR CalcuJ-ations for INTRA]2IN
LeveL L Benchmarks, EIR internal_ report, TM-45-92-34 (J2/7/52)
and Cotrigenda 21/9/82.
/12/ F. RöseJ and 'J. Hadermann, INTRACOIN Level I Calcujatìons withcode RANCHN, EIR internal- repott, TM-45-83-9 (7/2/92).
TABLE 7.1
BENCHIIARX CAST 1
COIIPAR ISON BETIIEENCALCULAIIOT{S FOR THE
23
LAYER 1
EXACT, RANCHN ÂND RÁNCHTIRST NUCLIDÊ
PARAl{ETERS : (IlrT?rE1¡81¡LlrP2rRl)
r*E -----::::::111I:1I:-:::-:::Ï13::-:il-:--::::---- - : : I : : - - - --- - : 1 ï : : - - -- : :: i 1 l-- - -1 I r : - - - - - T : !':'- -
22000. 5.181-1 ¿ 5.?A?-12 .5¿ 5.53?-1? ó.7924000. 2.15ó-11 2.161-11 .24 ?'?58-11 t"TZ2ó000. 7.190-1 1 7.198-11 .1? 7.t!1-11 3.35?g000. 2.014-10 2.015-10 .05 e.Gó3-10 ?.4530000 . tt.905-10 4.9tió-10 .03 4.995-10 1.8432000 . 1.0óó-09 t.0ó7-a9 .0? 1.081 -09 1 .373400c . ?.1 1 1 -09 e.1 1 2-09 .01 ¿.1 34-09 1 .073ó000. 3.8ó7-09 3.8ó7-09 .00 3.ß99-09 .8438000. ó. ó 30-09 ó. ó30-09 .00 ó.ó73-09 .óó40000. 1.c75-08 1.075-08 .00 1.Û80-08 .4842000. 1.óó1-08 1.óó1-08 .00 1.óó8-08 .4344CI00 . ?.4ó?-08 ?.46?-08 .00 ?.47 1-08 .351ó000. 3,5?1-08 3.521-08 -.00 3.531-08 .?848000. 4.E?8-08 4.878-08 .00 4.889-08 .¿250000. ó.573-08 é.5?3-08 -.00 ó.58ó-08 .195?000. 8.ó4?-08 8.ó42-08 .CIo 8.é55-08 -1554000. 1.11?-a? 1.112-07 .00 1.113-07 .135ó000. 1 .4oz-a7 1 .407-07 .00 f .404-07 .1458000. 1.738-07 1.7f8-07 .00 1.739-07 .l)7ó0000 . ?.1 ?0-c7 2.1?0-07 .00 z,-1?7'97 .CI9
ó2000 . ?.550-07 ?,550-07 .00 ?,55?-A7 '08óå000. 3.028-07 3.028-0? .00 3.029-07 .05óó000. 3.553-07 3.553-07 .00 3.554-07 .03ó8000. 4.1?5-a? 4.1?5-0? .00 4.1¿6-A7 .0370000. 4.?4?-07 4.74?-97 .00 4.744-A7 ,447?000. 5.4a4-a7 5.404-07 .00 5.405-07 .0374000. ó.107-07 ó.107'07 .00 ó.108-07 .A?7ó000. ó.849-07 ó.849-e7 .00 ó.850-07 .0178000. 7.ó28-0? ?.6?'8-07 .00 7'ó30-07 '0280000. 8.441-07 8.441-07 a00 &.143-07 ,478?000. 9.?8ó-07 9,?8ó-07 .00 9.?87-Q7 '0e84000. 1.01ó-0ó 1.01ó-0ó .00 1.01ó-0ó .028ó000. 1 .1 05-0ó 1 .1 05-0ó .00 I -1 0ó-0Ó "0598000. 1.197-a6 1.197-0ó .00 1.f97-0ó -.0390000. 1.291-0ó 1.291-0ó .00 1.791-0ó .019?000. I .38ó-0ó 1 .3Eó-0ó . t0 I .38ó-0ó -' 0194000. 1.482-0ó 1.482-0ó .00 1.493-có .039ó000. 1,580-0ó 1.580-0ó .00 1.580-0ó .019g000. 1.ó78-0ó I .ó78-0ó .00 I 'ó78-06 .01
1 00000. I .776-06 1 .?76'96 ,00 I . 77ó-0ó -.01102000 . 1.875-0ó 1.875-0ó .00 1.879-0ó .0?1 04000.1 0ó000.'t 08000.1 I 0000.I 1 2000.t I 4000.11óCI00,
1.973-0óe.071 -062.1óE-0ó¿.265-06z . 3 ó0-0ó2.454-0ó?.547-t6
I .973-0ó?.a7 1 -0ó7.1 6 8-0ó2.?6 5-0ó2.3ó0-0ó? .151-462.5t,7-û6
.00
.00
.00
.00
.00
.00
.00
1 .973-0ó?.a7 I -0ó2.168-0ó2.?64-46¿.3ó0-0ó?.15 1 -0ó2.547-06
010100a?000000
a
a
a
3
a
-24-
TABLE 7 .1 ( CONT. )
BENCHf{ARK CASE 1 LÂYER 1
c0¡TPARIS0N BETTJEEN EXACT, RAHCHN ÂNDCALCULÅTIONS FOR IHE FIRST NUCLI DE
PARAFIETERS : (11rTZ rBl rEl rLl rp¿ rRl )
RÂHCH
TIIíECONCENTRÂTIONS AND DEVIATIONS TOR 3 E34-U
RAICCHN DEV(Z) RANCH DEV(U)EXACT
1 1 9000.1 ?0000.I 2?000,1¿4000.I ¿ó000.128CIû0.1 30000.1 3 2000,1 34000.1 3ó000.138000.1 40000.142000.144000.I 4óû00.1 49000.1 50000.1 5e000,I 54000.1 5ô000.158000.1 ó0CI00.1 óU 000.1ó40û0.1 óó000.1 ó8000.1 70000.I 7a000.I 74000.1 7ó000.I 78000.1 80000.1 8¿000.1 84000.1 8ô000.1 88000.1 90000.1 9a000.f 94000.I 9ó000.1 9E000.200000.2û2000.204000.¿0ó000.208000.210000.212000.
2.ó38-0ó¿ . ? e8-0ó2.81é-0ó?.90?-ûó2.987-çJó3.0ó9-0ó3. 1 49-0ó3.2?6-463 .3 01 -0ó3. 373-0ó3.4tr2-063.508-0ó3 . 5 70-063. ó?8-0ó3.é82-A63.732-A63.776-0ó3.615-063 . I 49-0ó3.878-0ó3 .9 0f -063.918-0ó3.9 e9 -0ó3.935-0ó3.934-063.9e9-0ó3 .9 1 7-t63.901-063.I79-0ó3.g53-0ó-1.8?2-063 . 7Eó -0ó3.7 4 7-0ó3.703-0ó3.ó57-063 . ó0ó-0ó3.553-063.497-0ó3.439-0ó3.379-0ó3. 3 1 7-0ó3.¿55-0ó3. I 88-0ó3.1 ?1 -0ó3.054-062.9Eó-0ó?.917-A62.849-06
? . ó39-0ó2..728-0ó¿.8 1 ó-0óe.90e-0ó?.987-463 .0ó9-0ó3.1 49-0ó3 .2 2ó-0ó3.301 -0ó3.373-tó3.44U -0ó3,508-0ó3.570-0ó3. ó?g-0ó3 . ó82-CIó3.7 3a-0ó3 .77ó-0ó3.81 5-0ó3.849-0ó3 . I 7E-0ó3.901-0ó3 .91 8-0ó3.9¿9-0ó3.93 5-0ó3.934-0ó3.929-0ó3 .9 I 7-0ô3 .90 I -0ó3,879-0ó3.853-0ó3.8??-0ó3.78ó-0ó3 .7 47 -063.703-0ó3.ó57-463 . ó06-0ó3.553-0ó3 .497-0ó3 .4 39-0ó3.379-063.317-463.255-0ó3.1 8g-0ó3,121-0ó3 .054-0óe.98ó-0ó2.917-0ó2.849-0ó
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.0c
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00-.00
.00
.00
.00-.00
.00
.00
.00
.00
.00-.00
.00
2.ó38-0ó?.7?g-a62.81ó-0ó?.902-0óz .987 -463.0ó9-0ó3 .1 49-0ó3 . ? e6-063.301-0ó3.373-063.44¿-063.508-0ó3 .570-0ó3.óe8-eó3.ó8¿-0ó3.73 1 -0ó3.77ó-0ó3.E15-0ó3 .84 9-0ó3.878-063 .900-0ó3.9 1 7-0ó3.9e9-0ó3.934-0ó3.ç31-0é3 . I 28-0ó3.9 1 7-0ó3.901 -0ô3 .8 79-063.853-0ó3.8?Z-0ó3.78ó-0ó3 .7 47 -463.703-0ó3 . ó 5ó-0ó3 .60ó-0ó3.553-0ó3.497-0ó3.419-0ó3.179-063.3 1 7 -A63.253-0ó3.1 88-0ó3.1e1-0ó3.054-0ó? . 9 86-0ó2.917-46¿.848-0ó
-.0 ?-.01- .01- .01
.01
.01
.01-. t0
,00-.01-.01
.00- .01-.0f- .01-.4?
.00-.01- .01
.01-.01-.02
.00-.01-.01-. t?-.01
.00-. 01
.0û
.01-.01
.01-.01-. 01-.01-. 01-.0'l-.01
.00
.01,01.01
-. 01.00.00
-.01-. CIa
-25-
TABLE 7 .1 ( C0r{T. }
BENCHüARK CASE 1 LAYER 1
coirPARIs0N BEITJEE¡{ EXACIr RANCHN Al{DCALCULATIOiIS FOR ÏHE FIRST ITIUCLIDE
PARATqETERS : (¡lrTZrBlrEl;¡1 rPZrRl)
RANCH
lIl,lECONCENTRATIOHS A¡ID DEVIAÏIOI{S FOR 234-u
RANCHI*l DEV(Z) RAilCH DEV(l)EXACT
e14000.2f ó000.2 1 8000.z ?0000 .?2e000.¿?4000.2 2ó000 .2?8000.?30000.232000.?34000.¿3ó000.238000.e40000,24e000.?44000.a{ó000.a4E000.250000.¿5?000.254000.e5ó000.?58000.2ó0000.2ó2000.2ó4000.2 óó000 .2ó8000.e70000.?72000.274000.27ó000.278000.e80000.28 ?000 .284000.2Eó000.288000.290000.¿9e000.294000.?eó000,¿98000.300000.30?000.304000.30ó000.306000.
2 .779-06?.71A-462.642-46?.573-462.505-0ó? .437 -46?.370-0ó?.303-0ó?.2 38-0ó?.173-462.1 I 0-0ó?.047-461.985-0ó1 ,9?5-061.8ó5-0ó1.807-0ó1.750-0ó1 . ó94-0ót.ó40-0ó1 .5 87-0ó1.535-0ó1.484-0ó1.435-0óI .38ó-0ó1 .339-0ó1 .294-0ó1 .2 50-0óI . 2 0ó-0ó1.1ó5-0ó1.1?4-0ó1.084-0óI .04ó-0óI .009-0ó9.779-479. 380-079.04?-A78.71 5-078.398-07I . 092 -077.795-A77. 5 08-077 .231 -07ó .9ó 5-07ó.744-47ó .4 54-076.?1?-475.979-475.754-A7
2.779-467.710-0ó?.64 I -06e.573-062.505-0ó2.4t7-06?.370-0óe.303-0óe.238-f)óe . 1 73-0óe. 1 09-0ó2.047-061.985-0óI .9¿4-0ó1.8ó5-0óI .807-0ó1.750-0ó1 . ó94-0óI . ó¿0-GóI . 5 8ó-0ót .534-0ó1.4E4-0ó1.434-0ó1.38ó-0ó1.339-0ó1 . e94-0ó1 .249-061 . ?0ó-0ó1.164-061 .'l 24-0óI .084-0ó1.04ó-0ó1 .009-0ó9.7?7-A7I . 3 78-079.040-078.713-078.39ó-07I .089-077.79?-477 . 5 0ó-077.¿?8-07ó.9ó0-07ó.701-07ó.451-07ó. e09-075.976-475.751-07
2.779-06? "71 0-06?.64 1 -0óe.573-0ó2.505-0ó¿.437-062.370-0ó2 . 3 03-0óz . a 38-0ó2 ,1 73-0ó?.1 1 0-0ó¿.o47-0ó1 .98 5-061.9?5-0ó1.865-0óI .807-0ó1 .750-0ó1 . ó94-0ó1.ó40-0ó1 .587-0ó1.535-0ó1.484-0ó1 .434-0ó1,38ó-0ó1 .339-06I . ?94-0ó1 .2 5t-0ó1 . Z0ó-0ó1.1ó5-0óI .124-fìóI ,084-0ó1 .04ó-0ó1 .009-0ó9.7?9-A79.3E0-079.A42-O78.715-078.39E-078.092-077.795-A77. 5 08-077 .?3 I -076.9ô3-076.744-O76.454-076.¿.12.-CI75.979-475.754-47
- ,01- .01-. 0Z
.00
.02
.00
.01-.02
.00- .01
.02
.01-.01
.0?-.01.t0
-.01-.0u
.01r03.03.01
-.05-. 0?-.03
.0e
.04-.03
.04
.0¿-. 03
.0c
.01
.00-. 00-.00
.00-.00
.00
.00-.00
.00
.00
.00
.00-.00
.00
.00
.00
.00-.00-. 00
.00
.00
.00-.00
.00
.00-. 00-.00-. 01-.01
.00-. 01-. CIl-.0 f-.01-.01-.01-. 01-.01-. 01-.01-.01-.0e-.0?-.0e-.02-.0'f-. CI?
-.02-.42-.02-.02-.03-.03-.03-.03-. 05-.04-.04-.04-.04-.05-.05-.05
-26-
TABLE 7.1 ( COHT. )
BENCHTIîARK CASE 1 LAVER 1
c0ñPARIS0N EETyEEN EXACT, RÂNCHN ANDCALCUTAÎIONS FOR THE TIRSl NUCLIDE
PARAI.IETERS : (I1 rT2rBl rEl rLl,PU,Rl )
RANCH
ÏITAECOi{CENTRATTONS AND DEVIAT¡ONS fOR ; A}4-U
RAIìICHN DEV(Í) RA¡¡CH DEV(z)EXACT
I I 0000.31 2000.314000.31 ó000.51 8000.320000.322000.3¿4000.32ó000.3?8000.]30000.33¿000.314000.33ó000.338000.340000.34 ?000 .344000.34ó0e0.348000.3 50000.352000.354000.3 5ó000.358000.3ó0000.3óe000.3ó4000.3óó000.3óE000.370000.372000.374000,37ó000.378000.380000.36?000.384000.38ó000.388000.390000.39A000.394000.39ó000.398000.400000.40 e000 .404 000 .
5 . 5 3ó-075.3?7-475.1¿5-474.979-974 .7 41-474.5ótÌ-074.3E5-074.71 6-A74 .054-073 . I 97-073.7 46-473 . ó01 -073 .16 1 -073 .3 27-073.1 97-073.07A-072 .9 5 ?-072.836-07?.725-072,ó18-072.51r-472.41 ó-072.3e0-072.??9-47¿.144-47e.05ó-071 .97 4-471.89ó-07Í.8e0-071 .7 4E-07Ló7E-071.ó11-071.547-471.4E5-071 .42 6-07I .3ó8-071.314-071 . ?óf -071 .2 1 0-071.16?-971 .115-071 .070-071.027-479.8 5 5-089.4 58-089.07ó-088.709-08I .3 57-08
5.533-075.324-075.1?1-074.9?6-A74.738-474 .5 5 6-074.381 -074.21 3-074.050-073.894-073.7 4 l-073 . 597-0 73.457-473 . f,43-073. f 93-073 .068-07e.948-072.83 2-t7? .721 -0 7¿ .61 4-97?.51'f -07?,.t 12-47e.31ó-07¿.2?4-47?.1 5ó-07¿.051-07I .970-071.891-471 .81 ô-071 .7 43-471 .67 4-A71.6t7-A7l.rt U -071 .48 1-A71 .4? 1 -071.3ó4-071 .309-071.?57-471.?46-A71.157-071.111-071 .0óó-071.0?3-079.8 1 3-089.41 ó-089.034-088.ó68-088.31ô-08
-.0 ó-. 0ó-.0é-. 07-.07-.08-.09-.09-. 09-.09-.10-.11-.11-.12-.13-.13- .14-.14-.15-. 1ó-.1ó-.17-.18-. 19-.20-.20
I
a
I
I
a
222?2475262728?e313133343ó3739394143444ó4819
5.537-475.3?7-975.125-074.929-ø74.74t-074.5ó0-074.38 5-474.21 ó-074 ,05 4-073.897-073.7 47-073.é01-073.4ó1 -C73.3?7-073.197-473.97?-97e .95 2-A7a.83ó-07¿.7?5-07?.618-07?.51 5 -07?.416-472.320-07?.2?9-472.141-072 . Ò5ó-071.ç74-A71 .89ó-071.8¿0-071 .748-071.ó78-071.ó11-071.r47-071.485-07', .4?6-471.369-071 .3 1 4-471 .?6 I -071 .¿1 0-071.1ó2-071 .1 I 5-07L070-071.0?7-479 .8 5 5-089.458-089.07ó-08I .709-088 .3 57-08
.01
.00
.01-. 01-.00
.01
.00-. 00
.00-,01
.01-.01-.01
.01
.00-. 00
.00-.01
.00
.00
.01
.01-.0?
.02
.03
.0?- .01
.0e-.0 2
.01-.01-.01
.01
.00
.03
.0ó
.05
.01-.02
.03
.01
.00
.01-.00
.01
.00-.00-.00
a
a
a
a
a
a
a
I
a
a
-27-
T^BLE 7.1 (C0tlT.)
BEI{CHI{ARK CA SE 1 LA VE R 1
COIiPARISON BETUEEH EXACT ' RANCHN AND
CALCULATIO¡¡S FOR THE FIRST NUCLIDE
PARAI¡tETERS : (IlrT2rBlrElrLl rP2rRl)
RANCH
TII.IECONCENTRATIONS AND DEVIAT¡ONS FOR 254-U
EXACÏ RANCHT.¡ DEV (U ) R ANCH DEV(l)
40ó000.408000.4 1 0000,412000.414000.41ó000.418000.4 ?0000 ,4e?000.424000.4 eó000.429000.4 30000 .41e0CI0.43ó000.43ó000.4 58000.440000.44 ?000 .444000.44é000.44E000,450000.452000.454000.4 5óCI00.458000.4ó0000.4ó?000.4ó4000.4óó000.4óE000.470000.472000.¿+7ô000.47ó000.478000.480000.48e000.484000.466000.4EE000./r90000.49e000.494000.49ó000.498000.500000.
I .0 1 9-0E7 .69 1-087. 3 E3 -087.084-08ó.797-086.5? I -086,Ztó-08ó.002 -085.758-085 .5 24-085 . 3 00-085.084-084 . E 77-0E4 . ó79-084.488-084 .3 05-0E4.1 l0-083 .9ó1 -083 . t 00-083.ó45-083.49ó-083 .3 53-083.21ó-083.0E 5-0E?.959-082.838-08?.7?e-08?,61 0-082.504-082 .4 01 -08u.303-0E?.209-08e.1 1 E-0E? .03 I -081.948-0E1 .8ó8-08I .792-081 .71 8-081.ó48-081 . 5 80-061 .51 ó-081.453-081 .394-08I .337-081. ?8U -08f.22ç-081 .1 79-06I .1 30-08
7.ç78-487 .654-087.343-087.044-086.75 7-08ó.48?-086.¿1 7-085 .9ó4-085.720-085.487-083.262-485.047-CI84 .E4 1 -0E4.642-084.41 2-084.¿70-084. û95-083.927-083 .7óó-083.ó11-083.4ó3-083.320-083.184-083.053-08?,.927-08¿.807-08¿ . ó91 -08z . 5 80-08?.471-482.t7 a-06?.¿74-48?. 1 E0-08e.090-08a.004-081 .92 1 -08'l .842-08I .7óó-081.ó93-081.6¿3-CI81.55ó-081.491-08I .430-081 .370-0E1 .3 I 4-061 . e 59-081 . e07-081 .1 57-081.109-08
-. 51<?
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8.t19-087. ó95-087. 383-087.084-08ó.797-096 .5? 1 -08ó. e 57-08ó.003-085.759-085.525-085 . 3 00-085.,G84-084/.878-AE4 "ó79-084.488-084.305-084.1 30-083.9óZ-083.800-083. ô45-083.49ô-083.353-083.e1ó-083.085-.08¿.959-08? .838-082.7??-98¿ .61 1 -08u .504-082 .4A I -082 .303-082.209-08?.1 1 8-08e.03?-081 .948-081.8ó9-081.79?-081 .71 9-081.ó48-08I .580-081.51ó-0E1.454-08I .394-081.337-08I .28?-081.2¿9-081.179-08I .1 30-08
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28
lABLE 7 .1 ( C0t'lr. )
BENCHTqARK CASE 1
c0t¡tPARIs0t¡ BECALCULATIOI{S
PARA¡IETERS:
LAVER 1
T¡f EEN EXACT, RANCHN AI{D RANCHFOR THE FIRST NUCLIDE
( r l rT?,81, E1 ,L1 , P?, R1 )
TII{ÊCONCENTRATIONS A¡ID DEVIATIOIT{S IOR: E34-U
RANCHN DEV (T ) RANCH DEV(u)ÊXACT
502000.504000.50ó000.508000.510000.51e000.51 4000.51ó000.5 I 800t.5 20000.5e2000.52400CI.52ó000.5 eE000.530000.532000.534000.53ó000.5 38000 .540000.542000.544000.54ó000.548000.550000.55e0CI0.554û00.55ó000.5 58000 .5ó0000.5ó2000.5ó¿000.5óôCI00.5ó80fÌ0.5 70000.57e000.574000.57ó000.578000.580000.582000.584000.58ó000.588000.590000.59e000.594000.59ó000,
1 .0E4 -081 . CI39-08I . 9 ó7-099.558-099.1 ó5-09I . 7 89-098.428-09I .08U -097.750-CI97 .431 -097. î eó-09ó, I 33-09ô.55?-09ó. ¿83-09ó.025-095.777-995.539-095.31U-095. CIg3-094.884-094.ó83-094.491-094.30ó-094.1 29-093.959-093.79ó-093.ó40-093 .4 90-093 .3 47-093. ?09-091.077-092.95 1 -092 .829-09¿.71 3-09?.ó01-CIge .494 -092. 392-09¿.294-49¿.199-A9z . I 09-092 .0 e ?-491 .939-091.859-09I .783-091.709-091.ó59-091.57?-09t.507-09
1 .0ó3-081.019-089.7ó8-099.1óa-096 .974-09E.ó01 -098.e44-097.902-097.573-097.259-09ó .9 5 7-09ó . óó8-09ó .39 I -096.1¿5-095.870-095 . ó?ó-095 .39 ?-ûg5.1 ó7-091.95 ?-094 .71*6-494.548-094.359-094.177-t94.003-093.837-093.ó77-093.5 a3-093 .37ó-093 . ? 36-093.1 01 -09? .ç71 -49u "848-09¿.7¿9-A92.61 5-09?.50ó-092.40'l-09e.301-09e. 205 -09?.1 I 3-09¿.025-09f .940-091.859-091.781-091 .707-tg1.ó3ó-091.5ó7-09f.502-091.439-09
-1 .9?-f .9ó-2.01-2,05-u.09-?.1 4-?.18-?.¿3-¿.29-? r3¿-?.37-2.4?'? .47-Z.r?-7.5&-?.óe-7.67-2.7¿-?.77-¿.8?.-2.88-e. 93-2 .98-3 .04-3.09-3. 15-3.?1-3.2é-3.32-3.38-3 .44-¡.50-3 .5ó-3. ó?-3. ó8-3.74-3.80-3.8ó-3.93-J .99-4.0ó-4.1?-4.18-4.25-4.32-4.39-4.45-4.53
1.084-0EI .039-089 . 9ó8 -099.558-099. I óó-09E.7E9-098.4?8-098.08 e-097 .7 5 0-097 .43 1 -097.1?6-49ó .835-C9ó.552-09ó. u E3-996.025-095.777-A95 .5 4 0-095.312-095 .093-094.884-094.ó83-094.49 1 -094.30ó-094 . I 29-CI93 .9 t9-093.796-093 . ó40-093.491 -093.347-093 . e09-095 .077-09u.951-09¿.8e9-09?..711-09e.ó02-09?.195-992 .392 -09?.?94-Oqz . 1 99-09¿.1 09-092.4?2-491.939-091.859-09I .783-091 .709-091 .ó39-091.57?-AgI .507-09
.01-.04
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-.0a-. 01
.02-.01
-29-
TABLE 7.1 (C0¡¡T.)
BENCHI{ARK CASE 1 LAYER 1
COÍqPARISON BETUEEN FXACT, RANCHN ANDCALCULATTONS TOR THE IIRST NUCLIOE
PARAT{ETERS i ( I1 tT? rBl r E1 rLl rPer R1)
RANCH
CONCE¡¿TRATIONS AND DEVIATIONS FOR ?34-uTIIlE
EXACT RANCHN DEV(U) RANCH DEV(X}
598000.ó00000.ó02000.604000.ó0ó000.ó08000.ó10000.61 2000.614000.ó1 ó000.ó18000.ó20000.ó2a000.ó24000.ó ¿ó000 .ó28000.ó30000.ó32000.ó 34000 .63ó0CIo.ó38000.ó40000.ó4 e000 .ó44000.ó4ó000.ó48000.ó50000.ó52000.ó54000.ó 5ó000 .ó58000.óó0000.óó2000.óó4000.óóó000.óó8000.ó70000.ó7?000.ó74000,ó7óCI00,ó78000.ó80000.ó8 e000.ó84000.óEó0oCI.ó88000.ó90000.ó9U 000.
1.445-09I .38ó-091.329-091.?74-091.2??-491 .171 -t91 .1 ?3-091 .077-091.033-099.904-10g .497 -1 g9,107-108.73e-1 08. 373-1 08.0?9-1 07.ó99-107.383-107.080-10ó.789-1 0ó.510-106.?47,-145.9Eó-1 05.740-1 05.504-105. ?78-1 05.0ó1 -1 04 .8 54-1 04.ó54-104.4ó3-1 04. ?80-1 04.1 04-1 03.93ó-1 03 .77 4-143.ó19-103 .4 70-1 03.3eE-1 03.191-103.0ó0-1 02.935-10e.E14-102 . ó99-1 0?.58E-1 0?.48?-1 0e.580-t 0?.¿83-10?.189-10?.099-1 0? .01 3-1 0
1.379-091.3?1-091 .2óó-09f .2 1 3-091 .1ó2-CI9I .1 1 4-091.0ó7-091.0u ¿-099.796-',lO9.38ó-1 08.993-1 08.ó17-148. ?5ó-1 07 .911 -1 07.580-107.26?-14ó,95E-1 0ó.6ó7-1 0ó.388-1 0ó.1 20-1 05.8ó4-105.ó18-105.383-1 05.157-104.911-104.734-1 04.53ô-104 .34ó-1 04.1 ó3-1 03.9E9-103.8¿?-103. óó?- 1 03.508-103.3ó1-103.220-103 .08 5-1 CI
2.95ó-1 0¿ .83 ?-102,71 3-1 02,599-10?.490-10e.3Eó-1 02.28ó-î 02.1 90-1 02.098-1 02 .01 0-1 01.9?ó-101 .845-1 0
-4.59-4.óô-4.73-4.90-4.87-4.94-5 .01-5 .08-5.1ó-5. a3-5.30-5.39-5.45-5.53-5.ó0-5. ó8-5.75-5,83-5.91-5.99-ó .07-6.14-6.7?-ó.30-ó.39-6 .47-ó.55-ó. ó3-6.71-6.7q-ó.98-ó .9ó-7.05-7.13-7.¿¿-7.30-7.39-7.47-7.56-7.65-7.74-7.8¿-7.91-8.00-8.09-8.18-8.77-8.3ó
1.445-091 .38ó-091.3?g-091 .77 4-ø91.¿?Z-t91 .177,-091.121-09t .077-091.013-099.904-109.497-109.1 07-1 08.733-1 08.374-108.030-1 07.700-1 07. 383-1 07.0E0-f té.789-1 0ó.51CI-106.?43-105 .9 86-1 05.74C-105.504-1 05 .278-1 05 .0ó2-1 04 "854-1 04.ó54-104.4ó3-1 04.280-1 04.1 04-1 03.93ó-1 03.774-103.ó19-103.471-103.328-1 03.1 9¿-1 03.0ó1-10?.935-1 0?.81 5-1 0z . ó99-î 0e.588-1 02.487.-14z .380-1 0z.?83-f02.1 89-1 0e.099-1 02.A13-10
-.01.0e.0e
-.01.02.04
-.0 3-.01
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-. 00.CI0
-30-
TAELE 7.1 (CONT.}
BE}.¡CH¡TARK CASE 1 LAYER 1
COiIPARISON BETTEEN EXACT, RA}ICHN ANDCAtCUtATlONS FOR THE T IRST i{UCLIDE
PARAIIETERS : (I1t12rB1rE1rLlrPZrRl)
RA ¡iIC H
ITiIECONCEHTRATTOilS AND DEVIATTOI{S FOR 234-U
RAT.ICHN DEV (I } RAf\¡CH DEV(1)EXACT
ó94000.ó9ó0oCI.ó98CI00.700000.ZCI2000.704000.70ó000.7CIE000.710000.71 2000.71 4000.71 ó000.719000.72CI000.724000.724000.7?ó000.728000.730000.73 2000 .734000.73ó000.73E000,740000.742000.744000.74ó000.7å8000.750000.752000.754000.75ó000.758000.7ó0000.7óe000.7ó4000.7óó000.7ó8000.770000.772000.774000.7?ó000.778000.780000.78¿000.784000.78ó000.788000.
I .930-1 01.851-101.775-1A1.?03-101 .ó33-1 01 .5óó-1 01.502-10I .440-1 01.381-101.325-10I ' ?70-1 01 .218-101.fóE-101 .1 20-1 0I .075-t 01 .030-1 09.E82-119.47E-119.0E9-118.71 7-118.3ó0-1 I8.01 8-1 I7.69t-117.375-117.473-11ó.783-11ó.50ó-11ó.4f9-1 1
5.984-1 I5 .7¡9-1 1
5.504-115,279-115.0ó3-1 I4.E5ó-114.ó57-1 I4.467-114.284-114.f09-113.96 I -1 1
3.780-1 1
3. ó25-1 1
3.477-113.335-1 I3.1 99-1 I3 .0ó8-1 1
2.943-1 I?.82?-11?,747-11
1.767-141 .ó95-1 01.67,?-14I .554-1 01 .489-1 0I .42ó-1 0'l .3óó-î 0f .309-1 01 .254-1 01.e01-101.151-101.103-101.05ó-f01 .01 2-109.694-119.287-1 I8.697-1 1
8.524-1 1
8.1 6ó-î I7.E43-1 1
7.494-117.179-116 .877 -1 1
ó.5E9-1 1
ó.317-116.047-1 I5 .79¿-115.549-1'.|5.31 ó-1 1
5.092-1 I4 . E78-1 I4.673-114 .477-1 1
4.289-1 14 .1 09-1 1
f .93ó-11,.771-11l.ó1?-113.460-f I3.315-113.1 75-f 1
3.CI42-112..91 4-1 1
2.79 f -11z .67 4-1 1
? .5 ó?-1 I2.4J4-11?.351-11
-8.-8.-8.-8.-8.-8.-9.-9.-9.-9.-9.-9.-9.-9.-9.-9.-9r
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I .911 -1 0î E51-101 .775-1 01 .703-1 0I .633-1 û1,5ó6-10'l .50?-10I .440-1 01.381-101.3?5-10I .270-1 01.?18-101 .1 ó8-1 01 .1 20-1 0I .075-1 0I .030-'t 09.883-1 1
9 .4 7E-1 1
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2.747-11
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-iL -
TABLE 7.?
BEl{CHI,IARK CASE 1 LAYER 1
c0t{PAR I S0N BETTJEEH EXACT, RAiICHN ANDCALCULATIONS FOR THE TIRST NUCLIDE
PARAiTETERS : (If rT2r81rÉl rLl tPZrR?,
RANC}I
CONCET{TRATIOT,IS AND DEVIATIOT{S FOR 2!4-ttrIüE
EXACT RANCHN OEV (I) RANCH DEV(r)
ó000.1 2000,'t 8000.24000.30000.3ó000.4 e000.4E000.54000.ó0000,óó000.72000.79000.94000.90000.9ó000.
t 0e000.I 08000.114000.1 20000.1 2ó000.1 f?000.1 3E000.1 44000.1 50000.I 5ó000.1ó2000.1 ó8000.1 74000.1 80000.1 8ó000.1 92000.I 98000.204000.210000.21ó000.22¿000.22E000.234000.240000.24ó000.2 5?000.258000.2ó4000.e70000.z7ó000.?82000.288000.
5 .2 50-1 0?.429-471.583-0ó3.5E2-0ó5 . 3 7ó-0óó.ó5e-0ó7.443-067. EE I -0óE.087-0ó8.151-0ó9.1 30-068.0ó0-0ó7.9ó3-0ó7. E 5 0-0ó7 .731 -0ó7 . ó 07-067.483-0ó7.3 5 I -0óó.8 I 8-0ó5 .4 37-0ó3.811-0ó?.464-CI6I .51 8-0óç.079-075.333-073.09ó-071.785-071.A24-075.857-083.345-08I .90E-08I .08E-0Eó.20ó-093.541-092.0?1 -09I .1 54-09ó.595-103.771 -14¿.1 58-1 01 . Z3ó-1 07.0E7-114.0óó-1 1
?.334-111.341-1t7.712-124.4 38-1 22 .5 5 5-1?1.473-1?
5 .25 f -10?.4?9-47I . 5 83-0ó3.58?-0ó5.37ó-06ó.ó5¿-0ó7.k43-067.88 I -0ó8.087-0ó8.1 50-0ó8.1 ?9-0ó6.059-0ó7.9ó1 -0ó7.849-0ó7.7?9-0ó7. ó05-067.481 -0ó7.346-0óó.8 1 ó-0ó5.435-0ó3.E09-0óe .4ó?-0ó1.51ó-0ó9.058-075,51 7-073.078-071.769-071.011-075.750-085 .2ó2-081.84ó-081.043-085.888-093.319-091.8ó9-091.05U -095.919-103 .3 Z8-1 0I .871-1 01.05'l-105.905-1I5.31ó-111.8ó2-1 II .04ó- I 1
5.870-123.295-1 2I .859-1 2I .071 -1 e
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5 .4 ó7-1 02.413-471 .583-0ó3.58e-0ó5 .37ó-0óó.ó51-0ó7 .443-0ó7.E81-0ó8.087-0ó8.1 5 1 -0éI .1 30-0ó8.0ó0-0ó7.9ó5-0ó7. I 5 0-0ó7.711-467.647-067.483-0ó7 . f4 9-0óó .803-0ó5.411-0ó3 .78ó-0ó2 .44ó-0óI .50ó-069 .003-075.?87-CI73.0ó9-071.V69-47I .01 5-075.805-083.f1 5-081.891-0E1.079-08ó.151-093.509-092.003-091.144-09ó.53ó-1 03 .738-1 02 .1 39-1 01 .??5-1 07.A24-114.030-1 I2.31 4-1 1
1.3?9-117.644-1¿4.399-17?.533-1¿1.4ó0-12
4.13.1ó
-.01.01.01
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.01-.01-.00-.0e-.ZZ-.48-.ó5-.72-. 78-¡64-. Eó-.8I-. E8-.87-. E9-.89-.9 0-.85-.89-.89-.EE-.67-.89-. E9-.90-.91- .89-.88-.E7-.91-rE8-. E7-.87-.9 5
-32-
TABLE 7,3
BETTCHITîARK CASE 1 LAVER 1
corqPARI soN BETHEE¡{ EXACl, RANCHtTT ANDCALCULATIONS TOR THE FIRSl T{UCLI DE
PARAf'!ETERS : (IerTerBl rËl rLl rParRe)
RANCH
TIfiECONCENlRATIONS ÂND }EVTATIONS FOR 3 E45-Ciî
RÁNCH¡I DEV(U) RANCH DEV(U)EXACT
1 0000.1e000.1 4000.I ó000.19000.e0000.2200CI.e4000.eó000.e9000.30000.3 e000 .34000.3ó000.38000.40000.42000.44000.4 ó000 .4E000.50000.5 2000.54000.5ó000 "58000.ó0000.ó ¿0t)0 .ó4000.óó000.ó E000 .7CI000.7¿000.740CI0.7ó000.7E000.90000.8e000.84000.8ó000.E8000.90000.92000.94000.9ó000.96000.
I 00000.I 02000.I 04000.
2.34ó-08ó.ó1 e-081 . ?93-07e. t1 1 -07? ,687-A73.?3?-473.ó00-073,791-ø73.8?4-Q73.73?.-Or3.548-073.3 05-073.028-072.738-07?.449-A72.171 -AT1 .91 0-071.67 1-071 .4 5 5-071.eó1-071 .090-079.390-088.072-0Eó .9 24 -085.93r-085.073-084.354-083. ó99-083 .1 5 5-08? . é 89-08?.291 -08I .951 -0E1 . óó0-081 .41 3-08I . ¿02-081 .0??-088. ó90-097.5E8-09ó. ?80-095.33E-094 .537-093.855-093.27ó-09?.7 84-09?.5ó5-09e.01 0-091 .708-091.451-09
2 .34ó-08ó. ô I ?-08I .293-07u.011-07?.687-473.232-073 . ó00-0 73.791-A73.824-CI73,73?-073.548-073.305-073 .0e8-07?.738-472.449-87? .17 1 -071.910-071.ó71-071 .4 5 5-071.261-471.090-079.390-088.071 -08ó.924-085.911-085.073-084.334-083.ó99-083.155-08e.ó89-08?.?91-081.950-081.óó0-081 .4 I 2-09I .U 01 -081.0e?-088. ó89-097.3E7-09ó. e80-095.33E-094 .537-095.855-093 . e 7ó-09?.784-09e.365-092.0 I 0-09I .708-091.451-09
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.00
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.00
.00
.00
.00
.00
.00-.0 f
.00
.00
.00-.00-.00-.00-. 01-.01-.01-.01-.01-.01-. 01-.02-.0e-.01-.0 e-.0 f-.01-.01-.01-. 01-.00-. 01-. 00-.00
.00
.00-.01-.01
e .348-086.61 4-081 . ?93-07e.01 1-a72.ó88-073. ¿3e-073.ó01-073 . 791 -073.8e4-073.732-473.548-073.305-073.oeE-07?.738-47e.449-072.171-071 .91 0-071.ó71-071 .455-07I.2ó1-071 .090-079.590-088.07e-086.925-085.931-085.CI73-084.534-0EJ..70û-0I3.'t55-09?.ó89-08?.?9 1 -081.951-081.óó0-081.413-081.?02-08I .0u 2-0E8. ó90-097.388-09ó.280-095.338-094.517-093.85ó-093 .2 7ó-09?.784-49? .3ôó-092 .01 0-091.708-091.451-09
.10
.03
.03
.01
.02
.02
.01
.00- .00
.01-.01
.00-. 00
.CI1
.0e
.01-. CI?
-.02.01
-.02.0e
-.00.01.01.00.00
-. 00r01
-.01-.01
.00
.0?-.0e
.07
.02
.00
.00
.00-.00
.00
.01
.01-.01
.00,02.00.0?.01
33
TTBLE 7.3 (CONT.)
BENCH¡'ARK CASE 1 LAYER 1
ThIEEN CXACT, RANCHN AND RANCHFOR THE FIRST NUCLIDE
COIIPARISON BECÂLCULATIONS
PARÂËETERS: (t2 rT2rBl rEl rLl rP2 rR?)
TI iIEC0¡ICENTRATI0NS AND DEvIATI0NS F0R : ?45-Cfi
RANCHN DEY (T} RAiIC H DEV(x)EXACT
I 0ó000.1 09000.1 1 0000.I 1 ?000.I 1 4000.1 1 ó000.1 I 8000.1 20000.1 ¿e000.1 24000.1 ?ó000.1 ?8000.1 30000.132000.1 340CI0.I 3ó000.1 3E000.140000.142000.I 44000.1 4ó000.1 48000.1 500CI0.15?000.154000.I 5ó000.158000.I ó0000.I ó2000.I ó4000.1 óó000.I ó8000.
1.233-09I . CI4ó-09I .8 30-1 07.3ó9-1 06.050-104 .8 78-1 03.8óe-103 .009-1 CI
2.310-101 .75?-1t1 .31 5-1 09.7E3-117.??4-115.30?-1 1
3.870-11?.81 3-1 I2.437 -11I .4 70-1 1
1 .058-1 1
7.597-125.444-1?3.895-122,783-1 e1 .98 5-1¿1.415-1?1 .008-1 27.1 óE-1 35.09ó-1 53.ó21-15?.37?-13I .82ó-1 31 .?9 5-1 3
1.¿33-091.04ó-098.8e9-1 07.3ó8-1 0ó.049-1 04.87ô-1 03.8ó1-103.008-1 0?.310-101.75?-191.315-10g .779-1 1
7 .??.0-115.797-113.8óó-1 1
2.809-1 1
e.031-1 1
'l .467-111.05ó-1 I7.586-1?.5.43ó-1 e3.889-1 ¿7.778-121 .9E ?-1?1.61?-721.00ó-1 e7.151-135 . 08 ¿-133.óû9-13?.56?-131.817-131.288-13
-.01-.01-.01-. 01-.02- .02-.4?-.03-. 03-.03-.03-.04-.0ó-.08-.10-.13-.îó-.16-.15-. f 5-.15- .14-.15-.1ó-.17-.20-.23-.28-.33-.40-.48-.57
1.233-09I .0¿ó-099.817-107 .34ó-10ó .0u 0-1 04.844-1 03 .8 ¿9-1 0e.978-1 0?.284-1 01.730-101.?97-1t9.ó43-117.116-1 I5 .Z¿0-1 1
3.809-1 1
?.767-11? ,003-1 I1.445-1 1
I .040-1 I7 .465-1?5.349-1 e3.8¿6-1¿2.7t3-1?.1.950-1 ¿
I .389-f e9.89e-1 37.038-135 .003-1 33.555-132.5¿5-131.79?-131.27?-11
.0J
.00-.1 5
-.3 I-.50-.69-.8ó
-1 .Ce-1.13-1 .26-1.37-1.43-1.50-1 .54-1 .58-1.ó3-1.é5-1.ó9-1.ó9-1.73-1 .75-1.77-1.78-1 .79-1 .E3-1.92-1 .81-1.83-1.83-1.8?-1.84-1.81
-34-
TABLE 9.1
Benchmark Case 1
Parameters (I1, T2,81, E1, L1, P2, R1)
Nuc I ide Code cmax
Tmax
T TI I+
2 34-U
2 3 0-Th
226-Ra
RANCHR ANC HN
RANC HR ANC HN
RANC HR ANC HN
0.3935-050.3935-05
0.3468-070. 3468-07
0.6936-070.6936-07
1.65+51 .65+5
2 .28+52 .28+5
6. Z3+46.73+4
47+445+4
3.27 +53.27 +5
4+54+5
54+556+5
99
5.55.5
2 .28+5 9 .472. j0+5 9.65
4545
++
TABLE 9.2
Benchmark Case 1
Parameters (I1r T2,81, E1, L1, P2, R2)
Nucl ide Code c T TTmax max 7 7+
2 34-U
2 30-Th
226-Ra
RANCHR ANC HN
RANC HRANC HN
RANC HR ANC HN
152^05151-05
0.6215-060.6226-06
.2105-04
.2't 11-04
1.52+41.52+4
3. 12+43. 12+4
1 . 45+51.45+5
6. 1 0+46. 1 0+4
0. B0.8
3.73+53.73+5
3.323. 34
z. 83+42. B4+4
+5+5
1
1
00
1.31+5t.31+5
24+524+5
- 35 -
TABLE 9.3
Benchmark Case 1
Parameters (f2, 12, 81, 81, L1, P2, R1 )
Nuc 1 ide Codea)
max maxc T T T
7 +.7
245-Cn
237-Np
RANCHRANC HN
RANC HR ANC HN
RANCHRANCHN
1.43+51 . 43+5
.27 97 -05
.27 9 B-0 5
.3646-05
.3647 -05
].12+5J. 12+5
2.35+52. 16+5
7 .12+57 .12+5
6.36+56.36+5
00
00
00
00
233-u
a) RANCH results differ by 1% from those given in refs./10/l/11 / (result of inPut error)
TABLE 9.4
Benchmark Case 1
Parameters (12, T2, B1, E1, L1, P2, R2)
9.7 +49.7 +4
Nucl ide Code c T T T+.7max max 7
245-Cm
237-Np
RANCHRANCHN
RANC HRANC HN
RANCHRANCHN
0.3829-060.3825-06
.7787 -05
.77 93-05
.575 0-05
.5760-05
2 .55+42.50+4
1 . 40+51 .40+5
1 .12+51 .12+5
0g+4. 0g+4
3. 1 1+43. 1 0+4
00
00
6.35+46.35+4
2.63+52.63+5
4. 9B+44. 98+4
2.16+52.16+5
233-u
-36-
TABLE 10.1
Benchmark Case 2
Parameters (I 1, T2, 81, E1, R1 )
Nucl ide Code C T T Tmax max I +.f
234-U
2 3o-Th
226-Ra
RANC HRANC HN
RANC HRANC HN
RANC HRANC HN
0.5840-050 .5757 -05
0.4693-070.4637-07
77
1.55+51 . 5B+5
2.05+52.06+5
7.47,5
9.75+49. B0+4
9.75+41 .00+5
2. 49+52 .52+5
4. B6+54. 90+5
4. B6+54.92+5
+4+4
2.05+52. 0B+5
-0-0
0. 93850.927 1
TABLE
Benchmark Case 2
Parameters (I 1, T2, 81, E1, R2)
10.2
NucI id e Code cmax T
maxT T I+7
2 34-U
2 3 0-Th
226^Ra
RANCHRANC HN
RANC HRANCHN
RANC HRANC HN
0.8436-050.8471 -05
0.5684-060.5725-06
1 1 58-041191-04
1.25+51.26+5
1 .21+51 .22+5
2. 12+42.10+4
3. 45+43. 45+4
1.41+51.41+5
4 .21+54 .21 +5
4. 08+54. 1 5+5
5. 40+45. 20+4
00
33
08+406+4
-37-
TABLE
Benchmark Case 2
Parameters (f2, T2, 81, 81, R1)
10. 3
Nuc 1 ide Code C ,u* T ,r* T T7 +.7
2 45-Cm
2 37-Np
233-U
RANCHRANC HN
RANC H
RANCHN
RANCHRANCHN
o.5125-050. 4983 -05
0.5164-050.5092-05
2.89+52.92+5
2.03+52.05+5
1.67+51 .68+5
1 . 00+51 . 01+5
4.76+54. 86+5
4. 1 B+54. 28+5
0.00.0
TABLE 10.4
Benchmark Case 2
Parameters (1"2, T2, B1, 81, R2)
Nuclide Code cmax
Tmax
T-.r
T+.7
2 4 5-Cm
237-Np
233-U
RANCHRANCHN
RANCHRANCHN
RANCHRANCHN
0.1403-060.1534-06
0.9274-050.9281-05
0.6080-050.6104-05
3. B0+43. 80+4
45+545+5
1 .17+51.18+5
2. 08+42. 1Q+5
6. 1 B+46. 1 B+4
3. 48+43. 46+4
2 .21+52.21+5
l.87+51 .87 +5
7. 60+47. 50+4
-38-
TABLE 1 1. 1
Benchmark Case 1
Parameters (I1, T2,81, E1, L2, P2, R1)
Nuclide Code c T T Tmax max -.1 + I
234-U
2 3o-Th
226-Ra
RANCH
RANCH
RANC H
0.4980-7
0.7470-g
0. 1 494-g
8.12+5
8. 1 0+5
8.10+5
4.15+5
4. 1 6+5
4. 1 6+5
1 . 65+6
1 .65+6
1 .65+6
TABLE 11.2
Benchmark Case 1
Parameters (I 1, T2, B1, E1, L2, P2, R2)
Nucl ide Code cr"* Tmax T TI 7+
2 34-U
2 3o-Th
226-Ra
RANCH
RANC H
RANC H
0.1675-5
0. 15 12-6
0.4186-5
2.55+5
3.36+5
3.36+5
1.19+5
1.53+5
1.51+5
5.42+5
7 .32+5
7 .32+5
-39-
TABLE 1 1.3
Benchmark Case 1
Parameters (I2, T2, 81, E1, L2, P2, R1)
NucI ide Code
2 45-Cm
2 37-Np
233-U
RANCH
RANC H
RANCH
cmax
0.0
0.1433-6
0.3835-6
Tmax
2 .32+6
2. 04+6
T I
1 . 06+6
6. 86+5
T+.7
5. 44+6
5. 1 8+6
TABLE 1 1.4
Benchmark Case 1
Parameters (I2, T2, 81, E1, L2, P2, R2)
Nuc I ide Code cru* T^u* T 7T +.1
245-Cn
2 37-Np
233-tJ
RANC H
RANCH
RANCH
0. 1549-1 0
0. 8650-0 6
0.2257 -05
9. 50+4
f ,60+5
5.35+5
3.8 +4
3.54+5
1.67+5
1.61+5
1 . 84+6
1 . 63+6
234-
uR
AN
CH
RA
NC
HN
226-
Ra
23}-
Th
0(T
)
t .0:
06
l.ø-ø
7
I .ø
-øB
l.ø-ø
9
I .ø
- tø
I .0-
12
I ¡N a I
l.ø-ll
l.ø-1
32,
0+05
4.0+
Ø5
6.0+
05
T tY
I
FIG
UR
E 1
,1 C
ase 1
(11,
T2,
81,
El,
11,
P2,
R1)
ø,ø
0(T
)
1.0-
05
l.ø-0
6
| .ø
-ø7
I .0-
08
I .ø
-09
I .0-
l0
1.0-
ll
I .ø
- 12
L0-
t32,
ø+
ø5
4.ø
+ø
56.
ø+
05
T tY
l
234-
u226-
Ra
RA
NC
H
RA
NC
HN
I À l--J I
0.ø
FIG
UR
E 1
,2 C
ase 1
(11,
T2,
81,
El,
11,
P2,
R2)
I ¡N \) I
I .0-
G(T
)
I .0
-06
l.ø-0
7
t.ø-0
9 0g
l.ø-t
ø
l.ø-1
1
I .ø
- t2
I .0-
l32.
ø+
øS
4,0+
056 .0
+05
T
tY ¡
233-
U
237-
Np
RA
NC
H
RÂ
¡VC
¡üV
0.ø
FIG
UR
E 1
,5 C
ase 1
Q2,
T2,
81,
E1,
11,
P2,
Rl)
1.0-
c(T
) 06
1,0-
ø7
I .ø
-08
t .0-
09
l.ø-1
0
t .0-
t I
1,0-
12
t.ø-1
3
I lN (^) I
2.0+
054.
0+05
6 .0
+05
T
tY I
(12,23
7-N
pR
AN
CH
RA
NC
HN
233-
U
24S
-Cm
ø.ø
FIG
UR
E 1
,4 C
ase
IT
2,B
1E
l, 11
, P
2, R
2)
0(T
)
t .0-
06
I .0-
07
t .0-
08
I .0-
09
1.0-
10
L0-t
2
l.ø-1
3
t .0-
l I
2,0t
054,
0+08
230-
Th
234-
uR
AN
CH
RA
NC
HN
26-R
a
I À N I
0.tÐ
FIG
UR
E 2
,1 C
ase 2
(11,
T2,
81,
E1,
Rl)
6.0+
05
ï tY
¡
0(T
)
I .0-
05
I .0-
06
I .Q
-ø7
t .0-
08
t .0-
09
1.0-
10
I .ø
-12
t.0-1
3
I .0-
l I
z.Q
+ø
54.
0+05
234-
U
2 30
-Th
226-
Ra
RA
NC
H
RA
NC
HN
_t i J ¡ iJ ¡ I I =' a I IJ I
l À (n I
0.0
FIG
UR
T 2
,2 C
ase 2
(11,
T2,
81, E
l, R
2)
6.0+
05
T tY
¡
0(T
)
t .0-
06
| .0-
07
t .ø
-08
I .0-
09
I .0-
10
1,0-
12
1.0-
13
I .0-
t t
2 . C
+05
4 ,0
+05
6 .0
105
I .01
05 T
tY ¡
RA
NC
H
RåJ
VC
I'JV
37-N
p
233-
u
I À Ot I
0.c
FIG
UR
E 2
,3 C
ase 2
(12,
T2,
81,
El,
R1)
c(T
)
t .0-
06
1.0-
07
t .0-
e8
1.0-
09
t .0-
l0
1.0-
lt
l.Q-1
2
t .0-
t3I .0
+05
2.
0+08
3.
0105
4 .0
+05
T tY
l
37-N
p
RA
NC
H
RA
NC
HN
233-
U
245-
Cn
I A \ I
0.0
FIG
UR
E 2
,4 C
ase 2
(12,
T2,
81,
El,
R2)
226-
Ra
230-
Th23
4-u
RA
NC
H
z O
I À e I
I l- F- (J
55
5o
B1
T2
( r1
,
ø5
øs
5ø
TItl
E x
1ø5
øø
5ø
FIG
UR
E 3
,1 C
ase
1a
tl, L
2, P
2, R
l)
b z o CE
TE l- L) I l- t- L) CE
TIIl
E
FIG
UR
E 3
,2 C
ase 1
(11,
T2,
81,
El,
L2,
P2,
R2)
234-
U
230-
Th
RA
NC
H26
-Ra
I N \o I
2j7-
Np
233-
U
RA
NC
H
I Ul a I
z C] É F (J z o l- t- L) c
TIIl
E
FIG
UR
E 3
.3 C
ase 1
(12,
T2,
81,
El,
L2,
P2,
R1)
45-c
n
37-N
p233-
U
RA
NC
H
z o I ]-
tøø
TIM
E X
ld
FIG
UR
E 3
.4 C
ase 1
(12,
T2,
81,
El,
L2,
P2,
R2)
I (¡ FJ I
23)-
rh
coN
zRA
/RA
NC
H
226-
Ra
234-
u
z o H t- F (J (T
TIIl
E
FIG
UR
E 4
Cas
e I
(11,
T2,
82,
E1,
P2,
Rl)
I (Jr \) I
-53-
APPENDTX
Model- and Code Characterìzatíon
Codes (a) RANCH, (b) coNRA/aa¡,tcn
ItlodeL Characterization
A
7
a)
b)
.i ,# + ).n) co(x,t) = oi 'ð2
2CO(x,t) (x,t) +
+
"t* r*
¿dvr-dx
âx
.i ,# + \o) ck(x,t) = oì ð2 CO(x,t) C, (x,t)K2
ãx
*i-, ^o "n-r(*,r,
*{ u,*,+
- 2?2!-9-?22Y!e!i:222
one-dimensÍanal transpottTageted medium with piecewjse constant parameters
instantaneous lìnear sorption equiTíbríun
- Initial condìtíons
c (x,o): gk
- Boundarg conditíons
Upstteam
a) cO( 0 ,t)
b) Fk(t)
arbítrarg tíme dependent concentratíon at inTet
(oírichTet boundarg)
arbittarg plane source injection rate
ck(-*rt) 0 ìnfinite medíum
-54-
Downstream
CO(-rt) = 0 semi infínite medìum
- Restriction in patameters
<< J.
2. Code Charactetizatìon
- Name of code
RANCH, CONZRA
- Method of solutíon
Semí-analgtical solutíon: Laplace transformatíon
- Resfriction in tet vaTues
7
2. í < 70
3. ^'/1' . .4
- Size of Code and Progrannning Language
u'/7t
k<4
RArVC¡t..
CONZRA:
7000 program statements
800 program statements
CDC Fortran extended ETN 4.8
CDC Compass 3.6
B. COùE RANCHN
7. I{odel Characterization
-55-
2â
arcU(xrt) = f (xrt,ck)
ck(x,t)+sþf
¿2 ck(x,t) - 9(x,t,cl # co(xrt) +
ðx
+ h(xrt,Ck)
- Basic assumpt,Íons
one-dìmensional ttanspott
- Ínitial conditions
C(xrt=O) ang given function
- Boundarg condÍtions
upstream and. downstream boundatg condition of the form
(I
- Restriction
no restriction
2. Code Characterízatíon
- Name of code
R^A¡\TCIllV
- Method of solutíon
=Yx=xu
orx=xd
Spectral method in sPace
Gear method in t.ime
-56-
- Restriction in et val-ues
no resttiction
-2!r-"--2Í-c-o-9"--z2g-P-l2s-!2ry!]g-2zls_!tge_
800 ptogtam statements
CDC Fottran extended FTN 4.8
-57-
NomencLature fot Appendíx
Space and tíme dependencies of quantíties are not expTíciteTg given.
ia
c.K
ìD
F
F-R
f (x,trC. )K
g(xrt,CO)
h(x,t,CO)
j
k
_iI
x_d
ct' 3, \ô fx)
lo
Tongitudinal dìspersjon Tength þ]
concentratíon in water of nuclide k l"t/rt *ater]L-J
dispersion tensot lr2/fLJcross sectìon tot fTow of contaminated water at inlet
injection rate at geosphere inlet for nuclide k lcil{þ1
j*k
t
ang gíven functíon of x and t, whích mag also depend on
co(x,t) and cO_r(x,t)
index of Tager
nucLide ìndex
mìgration Tength in Tager i
retentìon. factor fot nuclìde k
tine (t = 0 at start of Teachíng)
interstitÍal water veTocitg lr/4
space coordinate fnl
cootdinate of the uf¡stteam b.c.
coordÍnate of the downstteam b.c.
constants fot the upstream and downstteam boundatg conditíons
Dìrac deTta function
decag constant of nucTíde k
t'liv
x
xu
7
t" l
