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1. Cycle no 1 incoming concentration to system 10 000000000000000 conc in this box 1 10 000000000000000 conc in this box conc in this box conc in this box conc in this box Cycle no 2 0 000000000000000E 000 0 000000000000000E 000 0 000000000000000E 000 0 000000000000000E 000 O1 B W N incoming concentration to system 0 000000000000000E 000 conc in this box conc in this box conc in this box conc in this box conc in this box Cycle no 3 1 0 000000000000000E 000 2 10 000000000000000 3 0 000000000000000E 000 4 0 000000000000000E 000 5 0 000000000000000E 000 incoming concentration to system 0 000000000000000E 000 conc in this box conc in this box conc in this box conc in this box conc in this box 1 0 000000000000000E 000 2 0 000000000000000E 000 3 10 000000000000000 4 0 000000000000000E 000 5 0 000000000000000E 000 NAGRA NTB 00 02 20 Cycle no 4 incoming concentration to system 0 000000000000000E 000 conc in this box 1 0 000000000000000E 000 conc in this box 2 0 000000000000000E 000 conc in this box 3 0 000000000000000E 000 conc in this box 4 10 000000000000000 conc in this box o 0 000000000000000E 000 Cycle no 5 incoming concentration to system 0 000000000000000E 000 conc in this box 1 0 000000000000000E 000 conc in this box 2 0 000000000000000E 000 conc in this box 3 0 000000000000000E 000 conc in this box 4 0 000000000000000E 000 conc i
2. 1661 54 X 2492 31 0 000007 0 000006 0 000005 X 3323 08 11076 92 21046 15 0 000004 Concentration kg m 0 000003 0 000002 0 000001 1 5E 05 6E 05 0 0001 0 00015 Distance m Fig 7 18 SANTA concentration profile for contaminant The corresponding chart for the analytical profile is shown below Analytical infinite half space 0 00001 0 000009 0 000008 E 0 000007 er 2 0 000006 NM c fx 2 0 000005 Se 2492 31 E 0 000004 3 3323 08 S 0 000003 11076 92 21046 15 0 000002 0 000001 0 1 5E 05 6E 05 0 0001 0 00015 Distance m Fig 7 19 Analytical profile This shows that the results generated by SANTA follow the analytical solution to an acceptable degree It must be noted that the analytical solution used above was for a pulse into an unbounded rock matrix whereas SANTA was calculating for a fixed boundary rock matrix 39 NAGRA NTB 00 02 Therefore the tail of the SANTA profile will be slightly at a higher concentration than that of the unbounded profile Fig 7 20 Also of note is the possible presence of any cumulative error associated with SANTA s implementation of sorption and diffusion SANTA Analytical Difference 0 000001 0 0000005 0 pu E 0 0000005 NM 830 77 0 000001 K 3323 08 5 21046 15 0 0000015 O O 0 000002 0 0000025 0 000003 1 5E 05 4 5E 05 7 4E 05 0 0001 0
3. Normalised concentration 0 2000 SANTA results 4000 6000 8000 Time years Fig 7 21 SANTA PICNIC breakthrough curve including advection dispersion and sorption 41 NAGRA NTB 00 02 Processes included in Fig 7 21 advection dispersion sorption B indicates results calculated using PICNIC version Il 0 20 Normalised concentration 0 2000 4000 6000 8000 Time years Fig 7 22 SANTA PICNIC B breakthrough curve including advection dispersion sorption and decay The processes included in Fig 7 22 advection dispersion sorption DECAY with a half life of 1000 years B indicates results calculated using PICNIC version II The output shows SANTA predicting the breakthrough curve almost identically to the PICNIC code prediction NAGRA NTB 00 02 42 8 Worked Example This section outlines the creation of a simple example file and then further modifies the input data to include more of the available options All the example files created in this chapter are also contained in the SANTA directory wrkex1 sin wrkex2 sin wrkex3 sin and wrkex4 sin for reference and will be installed automatically by the installation program 8 1 Worked example 1 advection Load SANTA by double clicking on the santa icon or start from the menu in Win95 NT The SANTA main screen will be displayed Click on the button labeled Create Load Update to access the input screen This is the main screen for entering the se
4. Now click on Use Values to return to the main screen In the section labeled Setup of run the text gt Main Channel Sorption should now be visible This section displays the basic setup of a run Again the text box to the right of the Start button will display the text Values successfully loaded Click on the Start button This time you notice that the displays indicating the concentration in the first and last box show a more smoothly fluctuating value they no longer rise and fall suddenly Once this has been completed click on the Grapher button Once in the Graphing screen click on the Plot button in the Distribution section This time there is still some tracer in the system although only a very small amount lt 1E 28 kg m NAGRA NTB 00 02 44 Next click on the Plot button in the Break thru section This illustrates the effect of sorption on the tracer as it passes through the system The blue line indicates the concentration level in the first box which rises more slowly to the value of the input source and then tails off when the input pulse is switched off The green line shows the break through in the final 45 box and shows the retardation of the pulse compared with the earlier example with no sorption and the smearing out and attenuation of the sharp step pulse caused by dispersion and sorption Click on the check box labeled Log Conc in the Break thru section and click on the plot
5. nni RE pl E EE HE B Simplified sketch of a water conducting fracture Simplified representation Direction of flow A Water conducting fracture B Matrix or hostrock alteration Aqueous Species Solid Species Simplified Box 3 D Representation Fig 2 1 A box derived from a representation of reality NAGRA NTB 00 02 A System Built up using basic boxes as building blocks Water Depth of matrix Rock Accessible Rock Matrix Main Channel Accessible Rock Matrix Accessible Rock Matrix Main Channel Accessible Rock Matrix D simplification of 3 Typical system constructed from boxes and comparative reality Fig 2 2 the rock matrix adjacent to a In addition to the boxes used to represent the main flow path fracture can be represented by additional boxes with different internal properties and volume diffusion from the fracture A typical setup with an advective channel and a diffusive rock matrix 1s shown in Fig 2 2 along with a simplification of a water flowing fracture Note For reasons of simplicity the rock matrix boxes are shown here with the same dimensions as the than the fracture to represent rock matrix which is accessible by main channel boxes n most cases the rock matrix boxes will have dimensions different than the main channel boxes NAGRA NTB 00 02 4 2 1 Overview of the SANT
6. and for example adjust the length of the flow path The conceptual shortcomings include the crude approximation of gradients for relatively large steps in space and time Also the processes of dispersion sorption and diffusion are computed sequentially rather than simultaneously Diffusion in the rock matrix is treated in one dimension without accounting for mass transfer between rows of boxes e g parallel to the fracture The implementation of combined dispersion and sorption is numerically exact only for the case of linear Freundlich type sorption Flow within the fracture is assumed to be represented by one dimensional flow through an equivalent homogeneous porous medium The numerical errors associated with the conceptual shortcomings have been assessed by calculations in comparison to exact analytical solutions or numerically more accurate models Errors associated with gradients are largest when gradients are largest as demonstrated in the case of diffusion form the fracture into the first box of the rock matrix during the first time step Subsequent time steps are modeled with sufficient accuracy such that incremental mass transfers are accurate for all but the initial time step of diffusion The sequential computation of the processes does not appear to induce significant error as demonstrated with a computation involving linear sorption in the rock matrix dispersion and radioactive decay Errors associated with non linear sorptio
7. 7 8 The function C u is required in this case to define the concentration profile present in the Rock Matrix that has been created prior to the main channel concentration dropping to 0 The resultant analytical concentration profiles are shown below in Fig 7 11 again for the period 1 to 2 hours Analytical Concentration Profile After Pulse 3 50E 05 3 00E 05 2 50E 05 2 00E 05 1 50E 05 1 00E 05 Concentration kg m 0 00E 00 0 0 00005 0 0001 0 00015 0 0002 Box number Fig 7 11 Analytical concentration profile after pulse Again a check was made on the mass flux through the Main Channel Rock Matrix interface As previously the code recorded the mass transfer but a different approach was taken to analytically calculate the flux Using Equation 7 8 to calculate the concentration at 80 separate points in each time period a more accurate set of concentration profiles was obtained With the aid of a small program and using the trapezoidal rule the area under each curve could be NAGRA NTB 00 02 32 approximated From this the mass of tracer in the Rock Matrix could be calculated Any change in the mass contained in the Rock Matrix between time periods would be due to a flux across the Main Channel Rock Matrix interface Thus the values of mass flux were calculated for all the time periods These results for the analytical and SANTA outputs are shown in Fig 7 12 SANTA Analytical Mass Transfer from
8. Morihiro Mihara JNC Tokai Japan Fiona Neall Neall Consulting Kunio Ota PNC Tono Geo science centre Japan and Paul Smith SAM Paul Smith kindly performed comparative calculations with the PICNIC code Thanks also to Laurence R Bently Department of Geology and Geophysics University of Calgary Canada for a detailed and thorough review of the code The Institute of Mineralogy and Petrology under the guidance of Tjerk Peters and the Rock Water Interaction Group at the University of Bern provided support and logistics throughout the code development 5 NAGRA NTB 00 02 12 References APPELO C A J amp POSTMA D 1993 Geochemistry Groundwater and Pollution A A Balkema Rotterdam Netherlands CARSLAW H S amp JAEGER J C 1959 Conduction of Heat in Solids 2nd ed Oxford University Press New York CRANK J McFARLANE N R NEWBY J C PATERSON G D amp PEDLEY J B 1981 Diffusion Processes in Environmental Systems The MacMillan Press Ltd London and Basingstoke NAGRA 19942 Kristallin I Safety Assessment Report Nagra Technical Report NTB 93 22 Nagra Wettingen Switzerland 1994 BARTEN W amp ROBINSON P C 1996 PICNIC A code to model migration of radio nuclides in fracture network systems with surrounding rock matrix Hydroinformatics 96 M ller ed Balkema Roterdam 541 548 A I NAGRA NTB 00 02 APPENDIX A Quick start run something now Follow these instructions afte
9. e L tat dans la derni re boite l autre extr mit de la fracture est enregistr et repr sente la courbe de restitu tion SANTA poss de un g n rateur de graphes incorpor pour la repr sentation des r sultats tels que des profils de concentration dans la fracture et la matrice rocheuse ou des courbes de restitu tion de mani re optionnelle les r sultats peuvent tre enregistr s dans un fichier Le contr le du programme de SANTA a t focalis sur la simulation de l coulement convectif de la diffusion dans la matrice rocheuse milieu limit ou non de l adsorption de la dispersion et de la combinaison de ces processus Ce contr le permet de comparer les r sultats de SANTA avec les solutions analytiques et les r sultats d autres mod les num riques Les tests comparatifs ont r v l une pr cision num rique suffisante pour les utilisations auxquelles SANTA est desti n e SANTA a t programm en combinant le FORTRAN avec le VISUAL BASIC en raison des capacit s math matiques prouv es du FORTRAN et de l environnement Windows convivial du VISUAL BASIC comprenant des graphiques int gr s et des facilit s d impression Le pro gramme fournit aussi un fichier d aide abondamment illustr ainsi qu un manuel et un guide de r f rence en ligne une s rie d exemples de fichiers d entr e est galement fournie SANTA fonctionne sur les environnements suivants Windows 3x Windows 95
10. then Flux pli ce Eq 4 2 where flux is defined from box I to box II which lie perpendicular to the main channel with concentrations C1 and C2 respectively and positive towards lower concentration NAGRA NTB 00 02 8 For the mass balance equation the mass transferred between boxes MT is defined MT Flux A Ot Eq 4 3 where A is the contact area area of rock matrix box parallel to main channel is the porosity of the medium in which diffusion occurs and f the time of one cycle To calculate the diffusion length the following equation was used 24 D t At time te 90 of the mass of the tracer will be contained in the matrix from depth 0 to depth Fig 4 2 Therefore the full depth of the rock matrix will be times the number of rock matrix boxes l NoSideBoxes Concentration 2 Sqrt Dt Distance Fig 4 2 Definition of diffusion length NOTE At present the code defaults to the depth of 10 rock matrix boxes This can be modified to more rock matrix boxes 1 e enable longer run times As the diffusion length is proportional to the root of time and the duration of the injection pulse 1s usually small compared to the duration of advective flow it might be possible to fix the maximum number of rock matrix boxes at a number 100 At present the maximum number of rock matrix boxes allowed by the code is 20 With the Flux pee and Mass Transfer MT DA tQ this gives _
11. D A o t C1 C2 2 JD t MT for a cycle Eq 4 4 9 NAGRA NTB 00 02 As all the factors except Cl and C2 will be constant throughout a run Equation 4 4 may be formulated as D A bt MT KDiff C1 C2 with KDiff Bor With the mass transfer known for a cycle the concentrations Cl and C2 can be updated accordingly Care must be taken when moving a mass of contaminant to or from a main channel box to a rock matrix box due to the possible different volumes of these boxes If CT is the concentration in the main channel box 1 and C2 is the concentration in the first rock matrix box I the updated concentrations can be calculated as follows With CI gt C2 C1 Cl MBWaterVolume KDiff C1 C2 Eq 4 5 MBWaterVolume C2 C2 RMWaterVolume KDiff Cl C2 Eq 4 6 RMWaterVolume where MBWaterVolume is the volume of water in the fracture box and RMWaterVolume is the volume of water in the rock matrix box In representing diffusion the code assumes that the initial incoming pulse to the first rock matrix box I off the first main channel box 1 can only travel the distance of one box per cycle Thus with a constant injection the tracer will reach the second rock matrix box II in the second cycle illustrated below Fig 4 3 with a depth of diffusion of 6 boxes This prevents a pulse reaching the furthest rock matrix box in the first cycle The code at this stage does not consider diffusion between diffe
12. Rock Matrix 2 50E 12 Analytical 2 00E 12 H SANTA Difference 2 1 50E 12 c je E c 8 1 00E 12 c O O 5 00E 13 0 00E 00 Time Hrs Fig 7 12 SANTA Analytical mass transfer from rock matrix As can be seen in Fig 7 12 the curve for the absolute difference labeled Difference between the analytical solution and the results output by SANTA follows the same pattern as before Again in the first time step after the pulse has passed through the largest concentration difference will be present As before the absolute error settles down and stabilises to a near straight line The error present when mass transfer occurs into the Main Channel will stay within known bounds and at acceptably low values 7 3 Sorption Dispersion The implementation of dispersion into SANTA is closely linked to retardation caused by sorption see Chapter 6 The testing for the effect of dispersion also double checks the implementation of linear sorption Thus for a fixed dispersivity set arbitrarily to 0 015 m a selection of Kd values Freundlich P 1 were used to investigate the combined effects of dispersion and linear sorption The Kd values used were 0 1 0 01 and 6 6E 3 and in order to see more clearly the development of the concentration profile the system was run for 8 cycles The code calculated automatically the number of main channel boxes required for the retardation caused by the different Kd val
13. back sub stitution into Equation 5 1 the value of C can be obtained 5 1 Diffusion with Sorption in the rock matrix Sorption is implemented in the rock matrix in exactly the same manner as for the main channel boxes outlined above An incoming tracer can react with the rock according to the Freundlich equation Equation 5 1 The only difference is that the tracer flow into or out of a box is NAGRA NTB 00 02 14 controlled by diffusion and not by advection as in the main channel In most common applications for rock matrix sorption the value for will remain at 1 0 due to the lack of experimental data to define The actual number and dimensions of the rock matrix boxes remain unaltered from those defined in Chapter 4 15 NAGRA NTB 00 02 6 Dispersion The effect of purposely induced numerical dispersion is used to implement physical dispersion in the main channel in combination with retardation due to sorption Up until now it is assumed that the entire contents of a box moves into the subsequent box during the time step of a cycle This conforms to the equation vAt Ax Eq 6 1 Where v is the average ground water velocity At the cycle time step and Ax the box length If this was the case each content of a box would be moved from one box to next with the box concentrations moving neatly with the box boundaries and remaining sharp plug flow The sharpness is blurred when the front transfer and box boundaries do not c
14. migration des tudes de sensibilit et l assistance au d veloppement de mod les conceptuels Les processus agissant sur un contaminant et pris en compte par SANTA comprennent la convection dans le conduit principal la dispersion dans le conduit principal l adsorption et la d sorption lin aires non lin aires dans le conduit principal la diffusion dans et de la matrice rocheuse l adsorption et la d sorption dans la matrice rocheuse la d sint gration radioactive et toute combinaison des processus mentionn s Le principe de SANTA est bas sur l utilisation d une s rie de boites identiques dans leurs di mensions physiques et leur contenu g ochimique pour repr senter un conduit principal homo g ne unidimensionnel fracture et de mani re optionnelle une matrice rocheuse homog ne unidimensionnelle accessible par diffusion cette matrice est dispos e d une mani re sym trique autour du conduit principal et repr sent e par des s ries de boites identiques Le transport dans le conduit principal est dict par le simple mouvement de la phase aqueuse d une boite la sui vante durant un laps de temps donn tandis que la diffusion dans la matrice rocheuse est r gie par une approximation discr te de la premi re loi de Frick Le terme source alimentant la premi re boite se limite une injection simple du contaminant de mani re continue et concentration fixe ou une impulsion de dur e limit
15. section labeled Setup of run will now display the text gt Main Channel Sorption gt Diffusion in Rock matrix and gt Rock matrix sorption should be visible Click on the Start button This run will take a little longer than the previous runs as SANTA is now calculating sorption in all of the rock matrix boxes in addition to the main channel boxes Once the run is completed click on the Grapher button to access the graphing screen The Distribution and Break thru graphs will be similar to the previous example the greatest changes having occurred in the rock matrix Click on the Plot button in the Diffusion section to access the diffusion screen Scroll through the graph to view the development of the diffusion profile This highlights the retarda tion caused by the sorption in the rock matrix In the previous example the diffusion pulse reaches further into the rock matrix in a shorter time load in the previous data perform a run and compare the results You have now created 4 example files with increasing complexity Explore SANTA further by changing parameters and settings in the saved files and plot the results Consult the user manual for further details on the running of the code NAGRA NTB 00 02 46 9 Concluding remarks on accuracy and limitations The limitations of SANTA are determined by restrictions on the geometry the limited choice of processes conceptual shortcomings and by limits to the numer
16. the problem of zero initial concentration and the surface at x maintained at a constant concentration C and a fixed barrier at x 0 The concentration profiles generated analytically for the unbounded half space Fig 7 5 and a bounded geometry Fig 7 6 and by SANTA Fig 7 7 for a select number of cycles after 0 05 12 05 and 23 45 hours are shown below SANTA output resembles that of the analytical solution to the bounded geometry Deviations between the infinite half space and the bounded geometry become increasingly larger with time Analytical Concentration profile Infinite half space 5 00E 05 amp 4 50E 05 4 00E 05 f 3 50E 05 D 3 00E 05 O E 2 50E 05 5 2 00E 05 e S 1 50E 05 1 00E 05 5 00E 06 0 00E 00 main 1 2 3 4 5 6 T 8 9 10 Box number Fig 7 5 Analytical concentration profile for the unbounded infinite half space after 0 05 12 05 and 23 45 hours 27 NAGRA NTB 00 02 Analytical Concentration Profile B Boundary at box number 10 5 00E 05 Rm 4 50E 05 4 00E 05 3 50E 05 3 00E 05 2 50E 05 2 00E 05 Concentration kg m 1 50E 05 1 00E 05 5 00E 06 0 00E 00 Box number Fig 7 6 Analytical concentration profile for a bounded system with a boundary after the 10 box after 0 05 12 05 and 23 45 hours SANTA Concentration profile 4 50E 05 4 00E 05 3 50E 05 3 00E 05 2 50E 05 2 00E 05 1 50E 05 1 00E 05 5 00E 06 0 00E 00 Concentration
17. 00013 Distance m Fig 7 20 Analytical difference 7 5 Sorption dispersion and radioactive decay This example file is based upon the Kristallin I reference case that is fully documented in NTB 93 22 NAGRA 1994 All data except the half life and retardation factor were taken from this report This reference case concentrates on the geosphere modeling region It describes the network of water conducting features which extend throughout the crystalline basement of Northern Switzerland The parameters defined below were used for input INPUT DATA The following input data is contained in the SANTA input file santex7 sin As a comparison the PICNIC code BARTEN amp ROBINSON 1996 was run with identical parameters SETUP PARAMETERS No of Boxes 10 Transit Time 159272 73 hours Duration of run 127418184 hours Duration of input pulse 521888954 391 hours Input Elemental concentration l kg m NAGRA NTB 00 02 MAIN CHANNEL Channel Width Path Length Fracture Aperture Fracture Porosity Density ROCK MATRIX No values used in this example SORPTION PARAMETERS Freundlich Alpha Freundlich Beta MISC SETTINGS Dispersivity Accuracy 40 lm 200 m 0 001 m 0 99 3000 kg m 5 m kg l 10 m 0 0001 In addition the half life for radioactive decay was set to 1000 years RESULTS FROM RUN The following graphs plot the output from SANTA along with the output from the PICNIC code for comparison
18. 3600x 5E 5 241E 12x 0 05x 3600 MT 1 677050983E 12kg Eq 7 1 Thus the updated values of concentration for the first Main Channel and Rock Matrix boxes are obtained using Equations 4 5 and 4 6 In this case KDiff C1 C2 1 677050983E 12 SE 5x9 89999984 E 6 1 677050983 E 12 Cl 4 983060091E 5 kg m Eq 7 2 9 89999984 E 6 and e 0 0x1 34164076E 7 1 677050983E 12 _ 1 25E 5 kg m Eq 73 1 34164076 E 7 NAGRA NTB 00 02 22 The code recorded to the special file Diff out the details for the first Main Channel box and its associated Rock Matrix boxes for every cycle Only the first few cycles and to a depth of only 5 Rock Matrix boxes are listed below The Mass transfer value is a running total of mass trans ferred into the Rock Matrix Also of note 1s the adherence to the pattern outlined in Fig 4 3 Main Channel Matrix 1 Matrix 2 Matrix 3 Matrix 4 Equations 7 2 and 7 3 Matrix 5 498306E 04 125000E 04 000000E 00 000000E 00 000000E 00 000000E 00 498729E 04 164063E 04 546875E 05 000000E 00 000000E 00 000000E 00 498862E 04 199707E 04 772705E 05 257568E 05 000000E 00 000000E 00 498983E 04 225403E 04 101425E 04 382347E 05 127449E 05 000000E 00 Mass Transfer cycle 1 1 677050999883405E 012 Equation 7 1 Mass Transfer cycle 2 2 934839228456907E 012 Mass Transfer cycle 3 4 061607854332973E 012 Mass Transfer cycle 4 5 068821086245884E 012 The fil
19. 6 9 L p7 Eq 6 9 Ax 2a Nob d elle je a Eq 6 10 PthL NN Ax Me Eq 6 11 With R defined as a virtual retardation factor that is used to define the numerical dispersion to simulate a desired physical dispersion a From this R value a virtual Kd can be derived and thus by using the existing code for sorption with this Kd value it is possible to induce the defined dispersion even when no effects of sorption are wanted by the user PthL Further u d e a Eq 6 12 R R 2a NoB PthL For stability R 21 i e NoB lt 20 As can be seen this method can also be combined with the user s defined sorption data in Equation 6 12 to produce a new modified Kd that includes a dispersion effect NAGRA NTB 00 02 18 7 Code Testing At present the code simulates advection diffusion dispersion and retardation by sorption As these aspects of solute transfer are fundamental in migration experiments it is essential that they can be proven to be working as they were defined in the model It was in order to demonstrate confidence in the basic mechanics of the code that the following testing strategies were carried out A set of test cases were constructed which enabled a close examination of any one or a combination of processes Due to the focus being on the mechanics of the code the input parameters might not necessarily represent any geological reality All results were plotted using Excel for direct comparis
20. A box model The geometry outlined above is restricted to the following elements e A 1 D homogeneous main channel fracture represented by an array of identical boxes both in dimension and content e A 1 D rock matrix optional accessable by diffusion arranged symmetrically about the main channel and represented by arrays of identical boxes both in dimension and content A single tracer source is input to the first main channel box The source can be 1 supplied continuously at a fixed initial concentration or 2 supplied as a pulse of finite duration at a fixed concentration The processes simulated by SANTA include e Advection in the main channel e Dispersion in the main channel e Sorption and desorption in the main channel e Diffusion into and out of the rock matrix e Sorption and desorption in the rock matrix e Radioactive decay e Any combination of the above processes 5 NAGRA NTB 00 02 3 Advection SANTA represents advection as the physical movement of a contaminant in this case using water as the transport medium In the very simple case of advection along a single flow path a linear sequence of identical boxes may be used Fig 3 1 Main Channel One cycle Input source Fig 3 1 simple advection system The SANTA code simulates the movement of water by moving the aqueous contents of box n to box n 1 Fig 3 2 The contents that were in the last box leave the system while the in
21. TA Br Analytical 1 6E 11 1 4E 11 1 2E 11 1E 11 Mass kg 8E 12 6E 12 4E 12 2E 12 0 2 0 8 0 9 0 3 0 7 lt LO e e 0 1 0 6 Time Hrs Fig 7 3 Cumulative mass transfer into the rock matrix at the end of each time step 0 05 hours produced with SANTA and analytical solution As can be seen the flux of mass calculated by the code is very similar to that predicted by the analytical solution A further analysis was made of the incremental mass transfer per time step This examined the calculated mass transfer per time step with that predicted by the analytical solution Fig 7 4 shows these values and the absolute difference between these results 29 NAGRA NTB 00 02 Mass Transfer per Cycle SANTA Analytical 4 01E 12 3 51E 12 Analytic Inc m SANTA Inc Difference 3 01E 12 2 51E 12 2 01E 12 Mass kg 1 51E 12 1 01E 12 5 1E 13 1E 14 0 05 0 15 0 25 0 35 0 45 0 55 0 65 0 75 0 85 0 95 Time Hrs Fig 7 4 Incremental mass transfer per time step calculated by SANTA and predicted by analytical solution Of particular interest in Fig 7 4 is the curve for the absolute difference labeled Difference between the analytical solution and the results output by SANTA Initially due to the starting conditions of maximum concentration in the main box and 0 concentration in the first side box the biggest error occurs This is due to the manner t
22. a contaminant while and D are known parameters whose values have been previously derived experimentally Therefore the concentrations in the water and rock can be determined with Equation 5 1 For most cases in natural environments is smaller than one NAGRA NTB 00 02 12 Initially an inventory of the mass of contaminant in the box 1s taken as follows INV WaterVolume C Rockmass C Eq 5 2 Replacing C INV WaterVolume C RockMass C Eq 5 3 It is now possible to solve for C There are three methods to solve Equation 5 3 depending on the value of J The first case is if B 1 producing a linear relationship between C and C For this case Kd and Equation 5 3 becomes INV WaterVolume C RockMass Kd C Eq 5 4 Making C the subject of the equation results in INV C anr el Eq 5 5 WaterVolume RockMass Kd Equation 5 5 is now immediately solvable as all the parameters are known By back substitution into Equation 5 1 the new value for C can be calculated The updated value for concentration in the water is the value that is now available to control diffusion into the rock matrix where the process of sorption occurs again The remaining methods are used if P z 1 In this case Equation 5 3 is non linear and requires a different approach One such approach is the use of the Newton Raphson one dimensional root finding method This method enables the honing in on the root or numerical sol
23. af IL n q r Q National Cooperative for the Disposal of Radioactive Waste TECHNICAL REPORT 00 02 SANTA Sensitivity Analysis of Nuclide Transport Aspects October 2002 D McKie and U Mader Hardstrasse 73 CH 5430 Wettingen Switzerland Telephone 41 56 437 11 11 E I Mm Q r Q National Cooperative for the Disposal of Radioactive Waste TECHNICAL REPORT 00 02 SANTA Sensitivity Analysis of Nuclide Transport Aspects October 2002 D McKie and U M der 1 DM Multimedia Ltd formerly University of Berne Switzerland 2 University of Berne Switzerland Hardstrasse 73 CH 5430 Wettingen Switzerland Telephone 41 56 437 11 11 This report was prepared on behalf of Nagra The viewpoints presented and conclusions reached are those of the author s and do not necessarily represent those of Nagra ISSN 1015 2636 Copyright 2002 by Nagra Wettingen Switzerland Allrights reserved All parts of this work are protected by copyright Any utilisation outwith the remit of the copyright law is unlawful and liable to prosecution This applies in particular to translations storage and processing in electronic systems and programs microfilms reproductions etc I NAGRA NTB 00 02 summary The SANTA Sensitivity Analysis of Nuclide Transport Aspects program is a simple box model created to simplify understanding of radionuclide or contaminant transport through a fractured medium SANTA is a teaching tool and allo
24. assed through and exited the system Next click on the Plot button in the Break thru section This may take a few moments as the code plots the aqueous concentration history for the first blue box and the last green box no 45 If desired any of the options marked Log Conc or Log Hours can be selected and the Plot button clicked again although in this case the differences will not be great later examples will highlight the usefulness of these buttons That 1s the simplest example of a run completed next we will add the process of sorption 8 2 Worked example 2 advection dispersion and sorption Return to the main window and again select Create Load Update The previously entered data will still be displayed In the section labeled Main Options in the top left of the window click on the check box titled Sorption in Main Channel The section labeled Sorption Parameters should now become active It was previously ghosted out and unavailable Leave the value for dispersion in the Main Options section adjacent to the Sorption in the main Channel check box at the default 0 015 m and enter into the Sorption Parameters section the following values Freundlich Alpha n kg 0 1 Freundlich Beta 1 Click on the Save button and save as testex2 sin This will save all the data to a file on your hard drive Remember to update your comments in the comments box this data is already contained in the file wrkex2 sin
25. at is a very quick start to using SANTA If you have a query at any point while running SANTA click on the Help button This will jump to the relevant section in the Help file
26. button in the same section This will now display the above graph but with a logarithmic scale for the concentration Now click on the check box labeled Log Hours and press the plot button again This will re plot the graph with the x axis in Logarithmic hours The user can now select the graph of their choice and with the use of the Save Graph button can save the output as a Windows Metafile This can then be directly imported to Microsoft Word or other pro gram and scaled without loss of resolution That 1s the example of a run with sorption dispersion completed next we will add the process of diffusion 8 3 Worked example 3 advection dispersion sorption and diffusion Return to the input screen and retaining the values already used click on the check box labeled Diffusion in Rock Matrix in the Main Options section in the top left of the screen This will now enable the section labeled Rock Matrix Leave the default value of 10 boxes for the Limit of Diffusion boxes and enter the following values Effective Diffusivity m s 1E 12 Matrix Porosity 0 05 Click on the Save button and save as testex3 sin This will save the updated data to a file on your hard drive Remember to update your comments in the comments box this data 1s already contained in the file wrkex3 sin Now click on Use Values to return to the main screen In the section labeled Setup of run the text gt Main Channel Sorption a
27. checked and proven FORTRAN code can be implemented within a Windows programmed front end These front ends are usually far more user friendly than a DOS prompt Another advantage 1s the ability to escape the 640 K memory limitation that programming in standard DOS dictates and the ability to use complex graphing routines and output directly to the printer The modular approach while using DLLs simplifies updating a program because one need replace only DLLs instead of replacing the entire program Thus when an updated or new method needs to be added the DLL containing the code need only be changed leaving the rest of the project unaffected DLLs can be accessed from any code capable of supporting DLLs thus any code present in a DLL can be utilised by other codes or Windows applications Packaging often used codes or routines into DLLs means that a selection of DLLs will be available for future programming projects NAGRA NTB 00 02 50 11 Acknowledgments This work s part of an extensive collaboration in model development supported by the Japanese Nuclear Fuel Development Cycle Institute JNC of Japan and the Swiss National Cooperative for the Disposal of Radioactive Waste Nagra The concepts involved in model development and the structure of code variants were initiated by Ian McKinley Nagra CH and developed under the management of Bernhard Schwyn Nagra CH with extensive input from Russell Alexander GGWW University of Bern CH
28. coming tracer can react with the rock according to the Freundlich equation thus retarding the transport of the contaminant when compared with purely diffusive flux To test the implementation of sorption in the Rock Matrix the code was set up as outlined below and the results recorded These were then compared with an analytical solution for the same process 37 NAGRA NTB 00 02 SETUP PARAMETERS No of Boxes 39 Transit Time 0 6 hour Duration of a run 8 86 hours Duration of input pulse 8 86 hours Input Elemental concentration E 5 kg m Dispersivity 0 015 Accuracy 0 01 MAIN CHANNEL Channel Width 0 3m Path Length 1 8m Aperture of Fracture 1E 4 m Fracture Porosity 0 99 SIDE MATRIX Effective Diffusivity 1 0E 12 Matrix porosity 0 05 Limit of Diffusion Boxes 10 SORPTION VALUES No values were entered for sorption in the main channel The option for sorption was switched off ADDITIONAL OPTIONS Rock Matrix Sorption Rock Matrix Alpha for sorption 0 0001 giving retardation factor of 6 7 Rock Matrix Beta for sorption 1 0 The above setup is stored in the file santex6 sin included on the installation disk When the above set of data was run SANTA output the following results Fig 7 18 for the development of contaminant concentration into the rock matrix at the different times shown All times are in seconds NAGRA NTB 00 02 38 SANTA output 0 00001 0 000009 0 000008 059 38 830 77 A
29. e after pulse 30 Fig 7 11 Analytical concentration profile after pulse 3l Fig 7 12 SANTA Analytical mass transfer from rock matrix 32 Fig 7 13 SANTA Analytical concentration profile for Kd 0 1 with 45 boxes 34 NAGRA NTB 00 02 VIII Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig 7 14 7 15 7 10 S ig 7 18 319 7 20 1 21 122 10 1 SANTA Analytical concentration profile for Kd 0 1 with 13 boxes SANTA Analytical concentration profile for Kd 6 6E 3 with 10 boxes SANTA Analytical concentration profile at 16 cycles Kd 0 1 SANTA Analytical concentration profile at 50 cycles Kd 0 1 SANTA concentration profile for contaminant 2 PAA ICAL DEO MI oneri tie Ho E taion Basel PMA VY UIC aL ICL CN Ce ansehen SANTA PICNIC breakthrough curve including advection dispersion and SOEDEIDIE Gn einer rs me eT tet OE Ide MI IK Ld ES SANTA PICNIC B breakthrough curve including advection dispersion SOEDLOH ad de nee a COMPATSOM OP codme methods e oo e Re o tn d etu d Em iate it I NAGRA NTB 00 02 1 Introduction SANTA Sensitivity Analysis of Nuclide Transport Aspects was created primarily as a teaching tool to simplify understanding of radionuclide transport through the geosphere and secondly as a simple scoping tool to quickly evaluate the effect of a range of retardation mecha
30. e Diff out including data for all cycles and Rock Matrix boxes off main channel box 1 produced the graph in Fig 7 1 for the time period 0 05 to 1 hour This shows the development of the concentration profile for each time step for the Rock Matrix adjacent to Box 1 of the Main Channel SANTA Concentration Profile 4 5 00E 02 5 00E 05 en 4 50E 05 k 1 50E 01 5 2 00E 01 Sues Kk 2 50E 01 m 3 50E 05 3 00E 01 o i 3 50E 01 D 3 00E 05 4 00E 01 C S 2 50E 05 a FE 5 00E 01 8 2 00E 05 5 50E 01 Q A 6 00E 01 O 1 50E 05 6 50E 01 1 00E 05 7 00E 01 7 50E 01 5 00E 06 ae 0 00E 00 _ 32 8 50E 01 0 0 00005 0 0001 0 00015 0 0002 0 00025 0 0003 9 00E 01 Distance m 9 50E 01 Fig 7 1 Development of SANTA concentration profile for the rock matrix adjacent to main channel box 1 time period 0 05 to 1 hour 23 NAGRA NTB 00 02 This profile can be checked directly against the analytical profile generated by using the following equation to define the concentration at a specific time f and distance x C t x C efl Eq 7 4 Equation 7 4 1s the 1 D analytical solution to the transient diffusion equation 2 Ficks 2 law Lid D 4 dt dx CRANK et al 1981 for the case of an infinite half space initially at 0 concentration with a fixed concentration C at x 0 CARSLAW amp JAEGER 1959 Using this equation a
31. e close matching of the SANTA results with those of the analytical solution for the case of an impermeable boundary The error present remains stable within known bounds and at acceptably low values The development of concentration profiles and fluxes become distinctly different for the bounded geometry 10 boxes and the infinite half space after about 10 hours For the SANTA calculations this is equivalent to 200 cycles which is equivalent to a characteristic diffusion depth of 4200 14 boxes The restriction of a bounded geometry imposed on SANTA can therefore be expected to mimic the infinite half space without significant error to at least twice the theoretical limit of the square of the number of Rock Matrix boxes 7 2 3 Diffusion into the Main Channel from the Rock Matrix The next step in investigating the codes handling of diffusion takes into account that a tracer can move out as well as into the Rock Matrix To investigate this the code was set up with the NAGRA NTB 00 02 30 previous values except that the run was for 2 hours Parameters shown below with the duration of the input pulse remaining at 1 hour This would mean that after 1 hour the main channel concentration in Box 1 would become 0 and so flow of tracer would be from the Rock Matrix into the Main Channel The equations that govern the mass transfer in the SANTA code are identical to those quoted previously except that in this case there will be a negative grad
32. ess of advection allowed to continue until the 15 cycle the predicted distribution profile would be as illustrated in Fig 3 5 This type of box model leads to a numerically accurate description of plug flow square pulse Normalised concentration 5 10 Box number Fig 3 5 Predicted distribution contaminant after 15 cycles 7 NAGRA NTB 00 02 4 Diffusion To represent rock matrix diffusion SANTA uses an array of boxes extending perpendicularly in the horizontal plane adjacent to the main channel This 1s illustrated in Fig 4 1 below with a depth of diffusion of 2 boxes and with the rock matrix boxes having the same proportions as the main channel boxes for clarity only 1 e this need not be so Diffusion in the main channel is omitted it is assumed that advection will dominate transport in the main channel In general the boxes of the rock matrix will usually be much smaller than the main channel boxes due to the shorter distance the diffusive flux can achieve within a cycle compared to the advective flow Main Channel Advective Flow 1 Input ie 2 Matrix Diffusion Main Channel Fig 4 1 Representation of diffusion in the rock matrix Diffusive transport into the rock matrix boxes is represented by Fick s first law CRANK et al 1981 Flux E D Eq 4 1 X where D is the coefficient of diffusion and a is the concentration C gradient in the x direction If this gradient is over a layer of thickness
33. et Windows NT V NAGRA NTB 00 02 List of Contents SUNMary re erc xL I Z sarnietfassup nissen ne Il SAR tu Det ou UU dU d DIN a MOREM eUUU E dU UDIN dad Coup IH E eM Contents RE m Um V WIS ROE REIP IN ER x nea eee VII 1 Introduction ei nn alt I The Box Model ona a ee ee tL CU cie icu 2 24 Overview of the SANTA box model 4 3 PUT uU M 5 4 Ditas oi e P 7 5 SOIPLOT serea a EN aes 11 5 1 Diffusion with Sorption in the rock matrix 13 6 DISBOESIUB ones sare Mei uat co tite ie ese pee MEE 15 7 Code Testing cenene e 18 7 1 INN eg Ly ON ee ee en ae ee ee 18 12 DIS ON ee lerne lese 20 72 1 Diffusion into the Rock Matrix from the Main Channel 20 2 2 Diffusion into the Rock Matrix with fixed boundary from the Main Channel scs ocu ee 25 7 2 3 Diffusion into the Main Channel from the Rock Matrix esses 29 1 3 SOPHO DISPENSON ee m eA MM EE ME 32 7 4 Dittuston wilb SOFDLOE ze ee ee 36 7 5 Sorption dispersion and radioactive decay ccccccccccececeeeeeeeeeeeeeesesseeeeeeeeeeeeeas 39 8 Worked Example nn Ev De en 42 8 1 Wotked example L 40VeCHOn he nas Le el 42 8 2 Worked example 2 advection dispersion and sorption cccccceeeeeeeeceeeeeeeeees 43 8 3 Worked example 3 advection dispersion sorption and diffusion 44 8 4 Worked example 4 advection dispersion sorption and diff
34. h the text Values successfully loaded displayed in the window to the right of the Start button in the Process Data section Press the Start button SANTA will now start to run While the calculations are being carried out the upper bar marked processed will fill up to show you how far the code has progressed in the calculations The bar marked Box 1 indicates the concentration of aqueous tracer 43 NAGRA NTB 00 02 existing at that precise moment in box 1 Similarly the bar marked Box 45 indicates the aqueous concentration for the 45 last box in the main channel When the calculations are completed the window to the right of the Start button will display the message Run Completed in xx xx xx where xx xx xx indicates the time taken for the calculations in hours minutes seconds this will vary depending on computer specification and complexity of the simulation It is now possible to access the Grapher button Click on this button to access the built in graphing window This window 1s divided into several sections At the moment only the sec tions labeled Distribution and Break thru are available There is at present no Diffusion data to plot First click on the Plot button in the Distribution section Nothing appears to happen It 1s actually plotting the aqueous concentration in all the main channel boxes and as shown there 1s no tracer in any of the boxes at the end of the run all the tracer has p
35. hat SANTA approximates the concentration d gradient a in Equation 4 1 by C7 C2 I After this initial error the absolute error settles x down and stabilises to a near straight line The error present will stay within known bounds and at acceptably low values 1 2 2 Diffusion into the Rock Matrix with fixed boundary from the Main Channel As can be seen from Fig s 7 1 and 7 2 the diffusion profile in the previous calculations does not significantly reach the furthest box in the Rock Matrix To investigate the effects of a fixed boundary at the end of box 10 the code was run with the parameters listed below The time of a run was extended to 26 hours 520 Cycles with a constant input during this time SETUP PARAMETERS No of Boxes 10 Transit Time 0 5 hours Duration of a run 26 hours 520 cycles 1 cycle 0 05 hours Duration of input pulse 26 hours Input Elemental concentration 5 0E 5 kg m Dispersion n Switched off Accuracy 0 01 Not used in this test NAGRA NTB 00 02 26 In this case due to the large number of cycles the code recorded data on only every 12 cycle Again the code output data for the concentration profile every 12 cycle This concentration profile can be compared directly with the analytical concentration profile generated using the following equation CARSLAW amp JAEGER 1959 c enler DER erd Ort DIE xt CA 1 leri JD ef 2 Dt Eq 7 6 giving the solution to
36. ical 0 000005 Analytical 0 1 0 000004 SANTA Kd 0 1 0 000003 0 000002 Concentration kg m 0 000001 0 o co lt N e 00 i N N 00 ph e 3 Y e S S e e S e Dimension m Fig 7 13 SANTA Analytical concentration profile for Kd 0 1 with 45 boxes Kd 0 01 SANTA vs Analytical 0 000005 Analytical 0 01 0 000004 BH SANTA Kd 0 01 0 000003 0 000002 Concentration kg m 0 000001 0 e co St CN oD co co l N N cO e e e e e e e Distance m Fig 7 14 SANTA Analytical concentration profile for Kd 0 1 with 13 boxes 35 NAGRA NTB 00 02 Kd 6 6E 3 SANTA vs Analytical 0 000005 Analytical 0 0066 0 000004 BH SANTA Kd 6 6E 3 0 000003 0 000002 Concentration kg m 0 000001 0 oO co t CN iS cO co st N N CO Dimension m Fig 7 15 SANTA Analytical concentration profile for Kd 6 6E 3 with 10 boxes Kd 0 1 at 16 cycles 768 s 0 000005 Analytical O 1 0 000004 HHM 16 cycles 0 000003 0 000002 Concentration kg m 0 000001 0 4 cO o N N co 0 48 0 56 0 64 0 72 0 8 Distance m Fig 7 16 SANTA Analytical concentration profile at 16 cycles Kd 0 1 NAGRA NTB 00 02 36 As a further test an investigation into the development of the concentration profile was examined at different time intervals The
37. ical accuracy These limitations render SANTA useful for the development of concepts scoping and exploratory calculations sensitivity studies and for teaching purposes SANTA was not developed as a tool for rigorous performance assessment The restrictions on geometry limit SANTA to calculations along a single flow path in a single fracture The model is essentially one dimensional with a pseudo dimension to represent interaction with a rock matrix adjacent to the flow path The major limitation on processes 1s the lack of generalised rock water interaction to account for the transient chemical evolution Sorption and desorption of a single solute is the only mechanism of retardation Sorption is of the Freundlich type either linear or non linear and can be defined individually for the fracture medium and the rock matrix The Freundlich model does not permit the definition of an upper limit for sorption capacity but the non linear option does allow for effects such as self sharpening fronts The combined equation implemented for advection dispersion and retardation requires that the length of the flow path the travel time and the number of boxes be no longer independent variables SANTA calculates the number of boxes required given the length of the flow path travel time dispersivity and sorption parameters for retardation The user should therefore avoid combinations of input parameters that result in either too few boxes or too many boxes
38. ient and hence an opposite direction of mass transfer SETUP PARAMETERS No of Boxes 10 Transit Time 0 5 hour Duration of a run 2 hours 40 cycles 1 cycle 0 05 hours Duration of input pulse 1 hour Input Elemental concentration 5 0E 5 kgm Dispersion n Switched off Accuracy 0 01 Not used in this test As the concentration profiles generated up to 1 hour are identical to those discussed in the earlier chapter these will be omitted from the graphs Therefore the time of interest 1s from 1 to 2 hours 10 to 20 cycle The output concentration profile generated by SANTA is shown below Fig 7 10 for the concentration in each of the 10 Rock Matrix boxes off the Main Channel box no 1 for each cycle SANTA Concentration Profile After Pulse 3 50E 05 3 00E 05 2 50E 05 E z 2 00E 05 o 1 50E 05 D O 6 G 1 00E 05 5 00E 06 0 00E 00 amp Box number Fig 7 10 SANTA concentration profile after pulse Again this can be checked directly against the analytical profile generated by using the following equation Eq 7 8 to define the concentration at a specific time t and distance x after a concentration profile has developed up to time t CARSLAW amp JAEGER 1959 31 NAGRA NTB 00 02 using Equation 7 4 e x C af setting t 1 hour oo x u x u l ETS x C u e 49 e 4Dt 24 ZDt 0 C x t where f t Eq
39. ine weitere M glichkeit besteht in der Ausgabe der Resultate in einem File Der Test des Computercodes richtet sich nach den in SANTA implementierten Prozessen wie advektivem Fluss Diffusion in die Gesteinsmatrix innerhalb eines begrenzten und unbegrenz ten Mediums Sorption Dispersion sowie einem kombinierten Prozesstest und vergleicht die Resultate von SANTA mit analytischen L sungsans tzen und Ergebnissen alternativer nume rischer Modelle Die vergleichenden Tests zeigen eine ausreichende numerische Genauigkeit f r die in SANTA vorgesehenen Anwendungen SANTA wurde in einer Kombination aus FORTRAN und VISUAL BASICS programmiert und vereint damit die bew hrten mathematischen F higkeiten von FORTRAN mit dem benutzer freundlichen Windows Programm VISUAL BASICS und all dessen integrierten grafischen und Druckm glichkeiten Das Programm weist ebenfalls ein vollst ndig erl uterndes Help File auf das on line ein Benutzer und Referenzhandbuch mit einer Reihe von Beispielen zu Eingabe Files enth lt SANTA erfordert folgende Betriebssysteme Windows 3 x Windows 95 und Windows NT IH NAGRA NTB 00 02 R sum Le programme SANTA Sensitivity Analysis of Nuclide Transport Aspects est un mod le sim ple de type boites successives destin la compr hension du transport de contaminants en mi lieu fractur Il s agit d un outil d enseignement permettant des calculs tendus l interpr tation d exp riences de
40. ion 6 2 PthL PthL Tt NoB Tt Nob E e HOD Pr NOD Eq 6 4 PthL PthL NoB NoB Eq 6 5 gt AB Eq 6 5 PthL l Eq 6 6 J D NoB m 1 4 Eq 6 7 24 R Therefore by introducing a dispersivity of to represent the actual field dispersivity it is now possible to determine the number of boxes NoD in the main channel necessary to produce this dispersivity using Equation 6 7 and hence travel time and the number of boxes cannot be chosen independently Addendum The above method is implemented in the code at present but has a simple limitation there must be some amount of sorption in the main channel in order to simulate any dispersion the number of boxes approaches 0 at 1 in Equation 6 7 A simple continuation of this theme results in a method to model dispersion when no sorption processes are defined in the main channel In the simplest case defined below just enough sorption to introduce a numerical dispersion equi valent to the physical dispersion is used and an increased average linear velocity u 1s used to offset the resultant retardation R Ax Following from Equation 6 1 u a At RR where u 1s the corrected migration velocity 1 e the effective velocity v is unchanged 17 NAGRA NTB 00 02 l Usi Ke Eq 6 8 vues 1 2a NoB Paper PIhL Which has been defined to have the following property l u a AX At Eq
41. kg m main 1 2 9 4 5 6 T 8 9 10 Box number Fig 7 7 Concentration profile calculated by SANTA after 0 05 12 05 and 23 45 hours NAGRA NTB 00 02 28 The mass transfer per 12 cycles was recorded by SANTA and this compared against the analytical results obtained from using the following equation CARSLAW amp JAEGER 1959 for the bounded geometry n F 1 M t 1C y 1 2 l e 4M I n erfc ZU n l n l nl Dt Eq 7 7 First the cumulative flux of mass across the Main Channel Rock Matrix interface was analysed In this case the analytical solution with a boundary Analytic B and SANTA data were plotted along with the analytical solution for the infinite half space Analytic A The results are plotted every 12 cycles and are shown in Fig 7 8 SANTA Analytical Cumulative Mass Transfer 9 00E 11 8 00E 11 7 00E 11 6 00E 11 9 5 00E 1 V S 400E 11 3 00E 11 Analytic A 2 00E 11 B Analytic B hN SANTA 1 00E 11 0 00E 00 LO LO D 2 2 2 2 L 22 2 2 2 2 2 2 2 2 LO 2 VD WO O N wv SO dO ON NT CS NYO DON SO do oO O NO ON 9 CN 2 TWO O lt lt N s v v v v v v v v CN CN CN CN Time Hrs Fig 7 8 Cumulative mass transfer across the main channel rock matrix interface plotted every 12 cycles Analytic A analytical solution for the infinite half space Analytic B analytical solution with a boundary after the 10 box and SANTA data Fig 7 8 shows clearl
42. llows the saving of graphs was created in Visual Basic The main reason for this approach was to retain the proven mathematical functions in FORTRAN that are familiar to most coders while adding the user friendly and familiar front end environment offered by Windows The basic component approach is outlined below along with the method employed in coding SANTA Fig 10 1 Traditional Coding Approach Executable Code DOS Command Line Component Approach Front End Compiler Linker Executable Code Subroutine Compiler Linker Compiler Linker Compiler Linker Windows Environment Comparison of coding methods 49 NAGRA NTB 00 02 Traditionally all subroutines were compiled into a single executable file exe An alternative approach is to use Dynamic Link Libraries DLLs DLLs are program modules that contain code data or resources that can be shared among Windows applications A DLL is basically the same as a Windows EXE file but one major difference s that a DLL is not an independently executable file although it may contain executable code A program can be constructed modularly using DLLs containing maths routines or other commonly called code The program while running loads in the relevant DLLs when required This 1s all done automatically and the end user need not be aware of the process The main advantage that SANTA gains by using DLLs is the way that existing and rigorously
43. lts can be output to a file Code testing focuses on SANTA s implementation of advective flow diffusion into the rock matrix within a bounded and unbounded medium sorption dispersion and a combined process test and compares SANTA s results with analytical solutions and results from alternative numerical models The comparative tests attest sufficient numerical accuracy for the purposes SANTA was designed for SANTA was programmed in a combination of FORTRAN and VISUAL BASIC retaining the proven mathematical ability of FORTRAN while including the user friendly Windows environ ment of VISUAL BASIC with integral graphing and printing capabilities The program also features a fully illustrated Help file containing an on line user manual and reference guide and comes with a selection of example input files SANTA runs on the following operating systems Windows 3 x Windows 95 and Windows NT NAGRA NTB 00 02 II Zusammenfassung Das Programm SANTA Sensitivity Analysis of Nuclide Transport Aspects ist ein einfaches Boxmodell das zum besseren Verst ndnis des Transports von Radionukliden oder Schadstoffen in einem gekl fteten Medium entwickelt wurde SANTA erm glicht als anschauliches Instru ment berschlagsberechnungen die Interpretation von Migrationsexperimenten sowie Sensiti vit tsstudien und ist bei der Entwicklung von konzeptuellen Modellen behilflich Dabei ber cksichtigt SANTA folgende Prozesse die auf einen Schadstoff ei
44. n calculations quickly and easily on a standard PC using the familiar environment offered by Microsoft Windows SANTA s integrated environment contains its own input files graphing abilities and output options thus enabling easy quick desk top scoping NAGRA NTB 00 02 2 2 The Box Model The aim of SANTA is to model the transport and retardation of a tracer in a fractured rock using a simple box model to represent the ground water flow path The basic principle of a box model is that this ground water flow path is divided into a number of identical compartments or boxes Fig 2 1 Each of these boxes 1s a simplified representation of the more complex reality with all boxes in the main fracture having identical dimensions and the same internal properties 1 e porosity rock density and volume A simple array of these boxes can be constructed to repre sent a water flowing fracture in one dimension and with an additional array of side boxes a rock matrix can be represented Fig 2 2 These boxes contain distinct phases namely the ground water and the rock with which it 1s in contact The water phase aqueous species 1s the mobile part and can move through the series of boxes carrying with it any tracer while the rock solid species is stationary and remains permanently in a given box retaining any retarded material A source supplies the first box of the fracture while the contents of the last box of the fracture leave the system NEN
45. n have not been quantified but it 1s expected that results are reasonably accurate as long as the non linearity 1s not extreme e g within the bounds of most experimental observations on rock material The implementation of dispersion by an equivalent amount of numerical dispersion is numerically accurate The distinct advantages of the simple box model are the non iterative and rapid computations the simplicity of the required input parameters and the transparent and modular implemen tation The comparative calculations presented in the previous section impressively demonstrate the capability of a rather simplistic approach 47 NAGRA NTB 00 02 The application of SANTA calculations to field problems is mostly limited by restrictions to the geometry and the simplified treatment of chemical processes Well defined portions of a natural system may however be explored and interpreted by SANTA simulations Also worst case scenarios for natural systems tend to minimize complexity and rely on simple concepts and hence might be explored with rather simple numerical models NAGRA NTB 00 02 48 10 SANTA was coded using a combination of FORTRAN and Visual Basic All the main numerical modules were coded in FORTRAN and checked first in a DOS environment This chapter outlines the method SANTA employs to combine FORTRAN code with a Windows Coding of SANTA front end or Graphical User Interface GUI The front end that controls SANTA and a
46. n this box 5 10 000000000000000 Cycle no 6 incoming concentration to system 0 000000000000000E 000 conc in this box 1 0 000000000000000E 000 conc in this box 2 0 000000000000000E 000 conc in this box 3 0 000000000000000E 000 conc in this box 4 0 000000000000000E 000 conc in this box o 0 000000000000000E 000 The computed results are identical to the expected results This confirms that the algorithm employed by the code to transport the contents of one box to another during a cycle without diffusion sorption or dispersion 1s performing correctly 7 2 Diffusion The testing of the representation of diffusion in and out of the rock matrix was broken down into several parts each of which examines a different aspect of diffusion The following tests look closely at the representation of diffusion into an unbounded and bounded medium with a continuous input or tracer supply and at the diffusion from this medium into the main channel when the input is switched off SANTA is treating the rock matrix accessable to diffusion as a bounded medium with an impermeable boundary set at a depth equivalent to the number of boxes considered for diffusion The impermeable boundary may be a feature wanted or unwan ted by the modeler The following tests attempt to establish the limit to which the de facto bounded geometry can mimic an unbounded geometry 7 2 1 Diffusion into the Rock Matrix from the Main Channel The
47. nd gt Diffusion in Rock Matrix should be visible Click on the Start button The displays indicating the concentration in the first and last box will rise similarly to the previous example but will fall much slower This is indicating the storage effect of the rock matrix retaining the tracer Click on the Grapher button to go to the graphing screen Click on the Plot button in the distribution section This shows the retardation effect of the rock matrix on the tracer pulse It is noticeably retarded compared to the previous example and at a higher concentration ranging from 3E 9 to 2E 11 kg m Click on the Log Conc and Log Hours check boxes in the Break thru section and then the Plot button This illustrates the effect of diffusion into the rock matrix with the retention of the tail caused by the effect of the tracer coming back out of the rock matrix 45 NAGRA NTB 00 02 For a better illustration of this change the Duration of run hours to 110 hours in the input screen and run again This will show the development of the tail over a longer time period remember to change this value back to 20 hours before continuing With diffusion now enabled it is possible to access the Plot button in the Diffusion section Click on this button to access the diffusion screen This section enables the user to see the con centration level in each of the rock matrix boxes adjacent to the main channel box chosen in the main
48. nisms on contaminant transport It 1s intended that SANTA s transparent and clear implementation of the processes will enable the user a basic insight into the calculations not afforded by more complex methods SANTA 1s designed to treat a single contaminant species without consideration of complex chemical interaction To achieve this SANTA employs the box modeling approach which very simply breaks down the ground water flow path into a linked series of identical boxes with the same physical properties 1 e porosity rock chemistry etc It is inside each of these boxes that the processes causing partitioning of the contaminant between the rock and ground water phases 1 e sorption can be simulated in a simple transparent manner The process of advection is implemented simply by moving the mobile ground water phase of a box into the next box down stream One main advantage of the box modeling approach 1s the clearly defined physical approach that results in an easily visualised system of boxes and their inter actions with each other The user can define all the relevant parameters used in the code These parameters are fixed physical dimensions and properties of the fracture material being modeled All other parameters are derived from these simple input parameters SANTA therefore does not need the input of a large amount of fit or constant of proportionality factors During the testing stages documented later any errors or limitatio
49. ns introduced by this box model approach are clearly shown This simple approach contrasts strongly with some of the alternative methods such as solving complex non linear sets of partial differential equations PDEs describing coupled reaction and transport in a multicomponent system by using finite difference techniques The methods to solve these sets of PDEs themselves introduce discretisation truncation and stability errors These errors are usually minimised by carefully selecting the grid spacing and time stepping to be used but without care an error term can quickly swamp the calculations This approach therefore requires the understanding of mathematical methods unfamiliar to many potential users of the code and as stated earlier it 1s the intention for the user to become involved in the actual methodology employed to solve the problem not simply to find a numerical solution to the problem It must be stated that in either case many of the values used for input terms are themselves subject to much debate and uncertainty Without a complete examination of the flow system a conceptual model is used for the internal properties of the system This limitation must be kept in mind when quoting the apparent accuracy of results produced by numerical codes in general To enable the user to avoid the long learning and familiarisation process necessary to use com plex models SANTA was coded to perform simple transport retardatio
50. nwirken Advek tion Dispersion lineare nicht lineare Sorption und Desorption jeweils entlang des bevorzug ten Transportwegs Diffusion in die und aus der Gesteinsmatrix Sorption und Desorption in der Gesteinsmatrix radioaktiver Zerfall sowie jede Kombination dieser Prozesse In SANTA wird eine Reihe identischer Boxen sowohl bez glich physikalischer Dimensionen als auch geochemischem Inhalt konzeptionell zur Darstellung eines eindimensionalen bevor zugten Transportwegs Kluft verwendet Als Option kann eine eindimensionale homogene Gesteinsmatrix gew hlt werden die f r Diffusion zug nglich und symmetrisch um die Kluft angeordnet ist und durch Reihen identischer Boxen dargestellt wird Transport in der Hauptkluft wird durch einfache Verschiebung der w ssrigen Phase von einer Box zur n chsten w hrend einer bestimmten Zeit durchgef hrt w hrend die Diffusion in die Gesteinsmatrix durch eine diskrete Ann herung an das Erste Fick sche Gesetz gesteuert wird Der Quellterm f r die erste Box beschr nkt sich auf die Zugabe eines einzigen Schadstoffs und erfolgt kontinuierlich mit einer festgelegten Konzentration oder als Puls einer bestimmten Dauer Der Zustand in der letzten Box am Ende der Kluft wird kontinuierlich erfasst und ent spricht der Durchbruchskurve SANTA verf gt ber ein integriertes Grafikprogramm um die Resultate als Konzentrationspro file Kluft und Gesteinsmatrix oder Durchbruchskurven darzustellen e
51. on with generated analytical results T 1 Advective flow To check that the advective process was operating correctly this first test was constructed which did not activate the routines for sorption dispersion or diffusion The aim was to follow an input pulse as it traveled through the system A set of test dimensions were selected to input a tracer pulse lasting exactly one cycle and watch this pulse for 6 cycles The dimensions used were as follows SETUP PARAMETERS No of Boxes 5 Transit Time 1 hour Duration of a run hour 12 minutes 1 2 hours 6 cycles Duration of input pulse 0 2 hours Input Elemental concentration 10 kg m Totally arbitrary number Dispersion n Switched off Accuracy 0 01 Not used by this version MAIN CHANNEL Path Length 5m Channel Width lm Aperture of Fracture lm Fracture Porosity 0 5 Not used by th s version These figures result in each box having unit dimensions ROCK MATRIX Rock matrix 1s not used in this test SORPTION VALUES switched off None of the values for sorption are used in this test 19 NAGRA NTB 00 02 With this setup it can be seen that the concentration in the boxes at each cycle 0 2 hours should be as follows Cycle 2 0 4 hours Cycle 3 0 6 hours Cycle 4 0 8 hours Cycle 5 1 0 hours Cycle 6 1 2 hours When the code was run the following data was output all concentrations in kg m TEST OUT DATA RUN re ADVECTION TESTS
52. orrespond vAt z Ax In this case the mixing of old and new concentrations in a box leads to a gradual smoothening of the transitions an effect termed numerical dispersion Numerical dispersion can be used to represent physical dispersion by choosing the cycle time At in relation to box length Ax in such a way that numerical dispersion is equal to the desired physical dispersion Physical dispersion is the result of uneven path lengths traveled by solutes in a porous medium combined with the effect of diffusion of solutes in the aqueous phase With this approach it is not possible to represent the process of dispersion by itself The method outlined here is closely linked with retardation caused by sorption Numerical dispersion can be calculated as follows APPELO amp POSTMA 1993 Ax vAt QQ Eq 6 2 R is the retardation factor and defined as m Pn where Pg the bulk density of the rock not including the pore spaces filled with water rock density 1 porosity Q the porosity of the main channel Kd Kd for sorption when 1 amp Kd Equation 5 1 NAGRA NTB 00 02 16 To determine the total number of boxes needed to represent a fixed dispersion value the following approach s taken Noting that PthL Tt PthL At and v2 NoB NoB Tt Ax where PthL NoB and Tt are the path length number of main channel boxes and transit time respectively Thus substituting back into Equat
53. put to the first box comes from the source term outside the system being either background water or an injected pollutant The process of moving water between boxes and subsequent solute redistribution 1s called a cycle a measurement of time used within SANTA Thus SANTA transports the water volume in each box to the next box in one of these cycles to simulate advection The time taken for one of these cycles 1s the residence time a solute remains in a box and is used in any calculations that are time dependent Initial Distribution V NN a Ep ES E ES Boxn Box n 1 Box n 2 Direction of advective flow Distribution after advection 1 cycle Ep E EG ES Boxn Box n 1 Box n 2 Note A leaves the system E is input to the system Fig 3 2 Advective flow NAGRA NTB 00 02 6 In SANTA s representation of advection the aqueous phase moves forward one box per cycle With a constant input of tracer and considering only advection no dispersion retardation etc the development of the predicted distribution profile 1s shown in Fig 3 3 after 5 cycles Normalised concentration 5 10 Box number Fig 3 3 Predicted distribution of contaminant after 5 cycles The predicted distribution profile is shown in Fig 3 4 after 10 cycles Normalised concentration 5 10 Box number Fig 3 4 Predicted distribution of contaminant after 10 cycles If the input of tracer is switched off after the 10 cycle and the proc
54. r installing SANTA to perform a run 1 Start SANTA by double clicking on the SANTA icon As SANTA holds a large amount of information in large arrays it 1s better to close any Programs you presently have running that also require a lot of memory 1 e Word Excel etc to prevent too much Hard drive activity as your machine uses virtual memory 2 From the main screen click on the Create Load Update button This will take you to the Input screen 3 From the input screen click on the Load button This will show a standard windows file requester 4 Select quick sin from the list shown this file should be included 5 Click on the Use Values button on the input screen This should return you to the main screen 6 Click on the Start button The program will start to run A progress meter will show the state of calculations and the other meters show the log concentration in the first and last boxes in the main channel 7 When the code has finished the message Run Completed will be shown in the text box next to the Start button 8 Click on the Grapher button This will display the Graphing screen Select any of the op tions you want and click on a Plot button Try looking at the diffusion profile Go back to the Input Screen and try running the same file but this time turn off the diffusion by clicking on the check box that says Diffusion in Rock Matrix and see the effect diffusion has on the system Th
55. rent rock matrix columns i e no diffusion occurs between any rock matrix boxes adjacent to Box 1 and any rock matrix boxes adjacent to Box 2 Diffusion occurs perpendicular to the main channel NAGRA NTB 00 02 Direction of advection 10 oO N e Z lt lt Cycle 1 I IM IV V VI Depth of diffusion 6 boxes oO N Cycle 2 I I IV V VI oO N gt Cycle 3 Al Direction of diffusion Fig 4 3 Consecutive build up diffusing contaminant in the rock matrix 11 NAGRA NTB 00 02 5 Sorption The process of sorption can occur in every box in the system and 1s the first process that SANTA calculates in a box on the incoming fluid followed by diffusion into the rock matrix if selected The concept underpinning the sequence of events relies on sorption being relatively fast compared to diffusion This is outlined in Fig 5 1 below with the possible order of events 1 Input Main Channel 2 Sorption 3 Matrix Diffusion 4 Sorption in rock matrix boxes Fig 5 1 Representation of sorption in the box model In the model sorption is described by the general Freundlich equation APPELO amp POSTMA 1993 C a C Eq 5 1 where C Concentration in the rock kg kg C Concentration in the water kg m QM Freundlich alpha for sorption m kg p Freundlich beta for sorption This equation describes the relationship between the concentration in the rock and the concentration in the water for
56. screen section Extra left at the default first box No 1 The main feature of this section is being able to scroll through the time periods recorded by using the scroll bar in real time Click on the right end of the scroll bar and hold the button down You will now see consecutive time periods illustrated forming an animation of the development of the tracer pulse into the system From cycle 22 0 782 hours onwards the input pulse has been switched off and the graph will now illustrate the tracer coming back out of the rock matrix this is analogous with Chap 7 2 3 Again the option exists to view the graph using Log concentration That 1s the example of a run with sorption dispersion diffusion and matrix diffusion completed next we will add the process of sorption in the rock matrix 8 4 Worked example 4 advection dispersion sorption and diffusion with sorption Return to the input screen and click on the check box labeled Sorption in Rock Matrix This option will only be possible if the previous Diffusion in Rock Matrix option has been selected Enter the following values Alpha for sorption 0 0001 Leave Beta for sorption at l Click on the Save button and save as testex4 sin This will save the updated data to a file on your hard drive Remember to update your comments in the comments box this data is already contained in the file wrkex4 sin Click on the Use Values button to return to the main screen The
57. set of values can be calculated using the same time steps as those used by SANTA and at distances equivalent to Rock Matrix boxes The graph obtained is shown in Fig 7 2 again from 0 05 to 1 hour As the scales used on both graphs are the same direct comparisons can be made Analytical Concentration Profile 9 00E 05 4 50E 05 4 00E 05 3 50E 05 3 00E 05 2 50E 05 2 00E 05 Concentration kg m 1 50E 05 1 00E 05 9 00E 06 0 00E 00 SZ ln ie 0 0 00005 0 0001 0 00015 0 0002 0 00025 0 0003 Distance m Fig 7 2 Development of analytical concentration profile for the rock matrix adjacent to main channel box time period 0 05 to 1 hour NAGRA NTB 00 02 24 While this shows the very close matching of the analytical results with those produced by the SANTA code a more informative comparison can be made by analysing the flux of mass across the Main Channel Rock Matrix interface per cycle The code already outputs a record of the mass transfer and this was plotted along with the analytical result generated from the following equation for time integrated flux CARSLAW amp JAEGER 1959 1 2 D C t A t x 0 dt Q hal x Jr Dt Eq 7 5 Fig 7 3 shows the cumulative mass transported into the Rock Matrix at the end of each time step 0 05 hours by the code and the analytical solution over the same time period Cumulative Mass Transfer SANTA Analytical 2E 11 1 8E 11 SAN
58. system was set up using the following parameters to investigate diffusion into the side matrix and the subsequent build up of tracer in each of the side boxes 21 NAGRA NTB 00 02 SETUP PARAMETERS No of Boxes 10 Transit Time 0 5 hours Duration of a run 1 hour 20 cycles 1 cycle 0 05 hours Duration of input pulse 1 hour Input Elemental concentration 5E 05 kg m Dispersion n Switched off Accuracy 0 01 MAIN CHANNEL Channel Width 0 5 m Path Length 2 0m Aperture of Fracture E 04 m Fracture Porosity 0 99 ROCK MATRIX Depth of rock matrix boxes 10 Value of effective Diffusivity 0 1E 11 m s Rock matrix zone porosity 0 05 SORPTION VALUES switched off None of the values for sorption are used for this test Sorption is switched off With these chosen values a constant input of 5E 05 kg m enters the first box With this con stant condition in box 1 the effect of diffusion could be analysed at every cycle every 0 05 hours The code writes to a special file the record of the concentration in each of the Rock Matrix boxes adjacent to the first Main Channel box Also included in this file was a running total of the transfer of mass through the Main Channel Rock Matrix boundary which enabled a check of the mass transfer per cycle to be made against the analytical results To calculate the mass entering the first Rock Matrix box on the first cycle Equation 4 4 1s used 1 12x0 1x0 05x0 05x
59. tup parameters of SANTA There are already a few fields filled with data These are default values and should be left as they are for the moment Starting with the Main Channel Geometry section enter the following numbers tip use the TAB button to move to the next field Channel Width m 0 5 Path Length m 1 8 Fracture Aperture m 0 0001 Fracture Porosity 0 99 Use the default value for Density which is already set at 2700 kgm In the Boxes and Cycles section enter the following No of Boxes 45 replace the value 10 Transit Time Hrs 1 6 Duration of run Hrs 20 Duration of input pulse Hrs 0 8 Input Elemental concentration kg m gt 0 00001 In the Misc Settings section leave the Accuracy setting at 0 0001 This 1s the relative error setting for the Newton Raphson solving algorithms used by SANTA If greater accuracy is required make this number smaller Note The smaller the Accuracy number the more calculations SANTA will have to make In the comments box enter a text description of the file In this case it is an example file using only the advection option Click on the Save button and save as testex1 sin This will save all the data to a file on your hard drive this data is already contained in the file wrkex1 sin Now click on the Use Values button This will return you to the Main screen again if any errors are present in data entry a warning will be displayed at this time wit
60. ues 33 NAGRA NTB 00 02 which meant that different run times had to be used to get to 8 cycles for each Kd value The set of input values used are listed below SETUP PARAMETERS No of Boxes 45 13 10 respectively for Kd values Transit Time 0 6 hours Duration of a run 0 11 0 4 0 5 hours Duration of input pulse 1 hour Input Elemental concentration 5E 6 kg m Dispersion n Switched off Accuracy 0 00001 MAIN CHANNEL Channel Width 0 5 m Path Length 1 8 m Aperture of Fracture 1E 4 m Fracture Porosity 0 99 ROCK MATRIX All of the values for the Rock Matrix are not used by this version The depth of diffusion is set to 0 boxes Diffusion is switched off SORPTION VALUES Value of Alpha for sorption 0 1 0 01 and 6 6E 3 respectively Value of Beta for sorption 1 0 Thus three runs were performed by SANTA for the three Kd values The outputs from SANTA were checked against the analytical result generated from the following equation APPELO amp POSTMA 1993 X C x t C 0 5 Co C erfc Eq 7 9 With the following boundary conditions 0 in this test case C x t C forx 0 t 0 C x t C forx 0 t 0 C x t C forx es t 0 input concentration 0 in this test case The results from SANTA were plotted along with the analytical results from Equation 7 9 These are illustrated below each recorded after 8 cycles see Fig 7 13 7 15 NAGRA NTB 00 02 34 Kd 0 1 SANTA vs Analyt
61. ur 22 Fig 7 2 Development of analytical concentration profile for the rock matrix adjacent to main channel box 1 time period 0 05 to 1 hour 23 Fig 7 3 Cumulative mass transfer into the rock matrix at the end of each time step 0 05 hours produced with SANTA and analytical solution 24 Fig 7 4 Incremental mass transfer per time step calculated by SANTA and predicted ON An yUlCa SO NICO i ceste ditio ta duoc etse een 25 Fig 7 5 Analytical concentration profile for the unbounded infinite half space after 0 05 12 05 and 23 45 BOUES ae 26 Fig 7 6 Analytical concentration profile for a bounded system with a boundary after the 10 box after 0 05 12 05 and 23 45 hours 27 Fig 7 7 Concentration profile calculated by SANTA after 0 05 12 05 and 23 25 hOUfs sodti hei doen ooo a des opis tip eR 27 Fig 7 8 Cumulative mass transfer across the main channel rock matrix interface plotted every 12 cycles Analytic A analytical solution for the infinite half space Analytic B analytical solution with a boundary after the 10 box and SANTA dali TTE 28 Fig 7 9 Mass transfer across the main channel rock matrix interface plotted every 12 cycles Analytic A analytical solution for the infinite half space Analytic B analytical solution with a boundary after the 10 box SANTA data and Difference between SANTA and Analytic B eese 29 Fig 7 10 SANTA concentration profil
62. usion with SOTPLION ee te estote inet ete bent di ocre ee D esters oti nd 45 9 Concluding remarks on accuracy and limitations 46 NAGRA NTB 00 02 VI 10 Coding of SANTIN G sense end Dubai vadit ae adu radius 48 11 Acknowlede melts u 50 12 References 51 APPENDIX A Quick start run something now A I VII NAGRA NTB 00 02 List of Figures Fig 2 1 A box derived from a representation of reality 2 Fig 2 2 Typical system constructed from boxes and comparative 3 D simplification OLTE esir eee ms sets 3 Fig 3 1 Sirple advectrOn system MN Nes 5 Fig 3 2 AVEC UNSS OMS ne ee nm en Sato I UU hd 5 Fig 3 3 Predicted distribution of contaminant after 5 cycles uueneeeseeeeeeeeeeeeeessssnesseneennnnnnenn 6 Fig 3 4 Predicted distribution of contaminant after 10 cycles ssessssss 6 Fig 3 5 Predicted distribution contaminant after 15 cycles 6 Fig 4 1 Representation of diffusion in the rock matrix cssssssssssneeeeneeneesesssnsnnnnnennnnnnnnnn 7 Fig 4 2 Definition or ditFusron Tenpllis so en 8 Fig 4 3 Consecutive build up diffusing contaminant in the rock matrix 10 Fig 5 1 Representation of sorption in the box model 11 Fig 7 1 Development of SANTA concentration profile for the rock matrix adjacent to main channel box 1 time period 0 05 to 1 ho
63. ution to Equation 5 5 to a desired accuracy Algebraically the method derives from the standard Taylor series expansion of a function in the neighbourhood of a point fa 6 2 fe FG 8 L8 o2 4 Eq 56 For small enough values of 6 and for well behaved functions the terms beyond the linear term can be ignored hence f x 6 0 implies fe with an overall error of f O f x 2 13 NAGRA NTB 00 02 This gives rise to the Newton Raphson formula where x initial guess and x improved estimate This Newton Raphson formula can now be used iteratively until some defined accuracy is obtained In the SANTA code this accuracy is input by the user If the relative accuracy is defined to be 0 01 then if x x S 0 01 X x the iterative process is halted and the value of C is taken to be x Re arranging Equation 5 1 to make C the subject C Eq 5 7 l Defining and B B Equation 5 4 becomes a INV WaterVolume A C RockMass C Eq 5 8 which 1s equivalent to f C WaterVolume A C RockMass C INV 0 Eq 5 9 The first derivative becomes f C B WaterVolume A C RockMass Eq 5 10 By putting B 1 in Equation 5 9 an initial estimate of C can be obtained Once this starting point and the desired accuracy are known a new value of C is solved iteratively by the Newton Raphson method outlined above Again with this new value of C known by
64. value of Kd was set at 0 1 using 45 boxes and SANTA produced output files at 16 and 50 cycles in addition to the previously recorded output at 8 cycles These were plotted with the analytical results and are shown in Fig 7 16 and 7 17 Kd 0 1 at 50 cycles 2400 s 0 000005 0 000004 0 000003 0 000002 Concentration kg m Analytical O 1 i 50 cycles co o N N co oO 0 000001 0 48 0 56 0 64 0 72 0 8 Distance m Fig 7 17 SANTA Analytical concentration profile at 50 cycles Kd 0 1 Errors are marginal and well within acceptable bounds in all test cases The input files to repeat the above runs are included on the installation disk santex1 sin santexlb sin santex1c sin santex2 sin santex3 sin santex4 sin santex5 sin The coupled implementation of dispersion and retardation sorption 1s numerically exact for linear sorption Results for weakly non linear sorption close to 1 are also expected to be within acceptable error bounds for scoping calculations Strongly non linear sorption B lt lt 1 8 gt gt 1 will still be modeled correctly in a qualitative way but results are expected to suffer from numerical imprecision 7 4 Diffusion with Sorption Sorption is implemented for the Rock Matrix in exactly the same way as the main channel boxes although in this case the transport of a contaminant 1s controlled by diffusion rather than advection An in
65. ws scoping calculations interpretation of migration experiments sensitivity studies and assists in the development of conceptual models The processes acting upon a contaminant explored by SANTA include advection in the main channel dispersion in the main channel linear non linear sorption and desorption in the main channel diffusion into and out of the rock matrix sorption and desorption in the rock matrix radioactive decay and any combination of the above processes Conceptually SANTA uses an array of identical boxes both in physical dimensions and geochemical content to represent a 1 D homogeneous main channel fracture with an optional 1 D homogeneous rock matrix accessed by diffusion arranged symmetrically about the main channel and represented by arrays of identical boxes Transport in the main channel is implemented by simple movement of the aqueous phase from one box to the next during a set time while diffusion into the rock matrix 1s governed by a discrete approximation of Fick s first law The source term supplying the first box is limited to a single contaminant input supplied conti nuously at a fixed concentration or as a pulse of finite duration The status in the last box at the end of the fracture 1s continuously recorded and represents the breakthrough curve SANTA contains a built in grapher to display results such as concentration profiles fracture and rock matrix or breakthrough curves or optionally resu
66. y that as time increases less mass 1s transferred into the Rock Matrix when a barrier is present after the 10 Box compared to the infinite half space The Rock Matrix is approaching equilibrium with the Main Channel box concentration when an impermeable boundary is present whereas for the case of the infinite half space diffusion into the Rock Matrix continues 29 NAGRA NTB 00 02 The mass transfer per 12 cycles for the analytical solution for the infinite half space the bounded geometry and SANTA is shown below Fig 7 9 The difference between the analytical solution with a boundary after the 10 box and the results from SANTA are labeled Difference SANTA Analytic B Mass Transfer per 12 cycles 1 00E 11 4 9 00E 12 Analytic A 8 00E 12 imB Analytic B 7 00E 12 SANTA PR 6 00E 12 Lr Difference 9 SANTA Analytic B 5 00E 12 gt 4 00E 12 3 00E 12 2 00E 12 1 00E 12 0 00E 00 I HIHI HT oh hel eh THT HTH Thee fT ES PET LO DD 2 22 2 2 2 2 2 on VO 2 2 2 2 2 2 LD LD VD W oO NY o GO Q COS N Te oN x O 0 st O O N Q O a 2 ONO 9 CN N s v v v v v v v v v N N N N Time Hrs Fig 7 9 Mass transfer across the main channel rock matr x interface plotted every 12 cycles Analytic A analytical solution for the infinite half space Analytic B analytical solution with a boundary after the 10 box SANTA data and Difference between SANTA and Analytic B Fig 7 9 shows th

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