Analysis of Some Laboratory Tracer Runs in Natural Fissures

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Tracer tests in natural fissures performed in the laboratory are analyzed by means of fitting two different models. In the experiments, sorbing and nonsorbing tracers were injected into a natural fissure running parallel to the axis of a drill core.
  WATER RESOURCES RESEARCH, VOL. 21, NO. 7, PAGES 951-958, JULY 1985 Analysis of Some Laboratory Tracer Runs in Natural Fissures LuIs MORENO AND IVARS NERETNIEKS Department of Chemical Engineering, oyal Institute of Technology, tockholm, weden TRYGGVE ERIKSEN Department of Nuclear Chemistry, Royal Institute of Technology, tockholm, weden Tracer tests n natural fissures performed n the laboratory are analyzed by means of fitting two different models. n the experiments, orbing and nonsorbing racers were injected nto a natural fissure running parallel to the axis of a drill core. The models ake into account advection, dispersion, diffusion into the rock matrix, and sorption onto the surface of the fissure and on the microfissures nside the matrix. For the second mechanism, one of the models considers hydrodynamic dispersion, while the other model assumes hanneling dispersion. The models take into account time delays n the inlet and outlet channels. The dispersion characteristics nd water residence ime were determined from the experiments with nonsorbing racers. Surface and volume sorption coefficients nd data on diffusion nto the rock matrix were determined or the sorbing tracers. The results are compared with values ndepen- dently determined n the laboratory. Good agreement was obtained using either model. When these models are used or prediction of tracer transport over larger distances, he results will depend on the model. The model with channeling dispersion will show a greater dispersion han the model with hydrodynamic ispersion, ssuming onstant dispersivity. INTRODUCTION The final disposal of radioactive wastes rom nuclear power plants has been studied n many countries. n Sweden, crys- talline rock has been selected as the most suitable bedrock in which to build a repository. f a canister s broken, adio- nuclides will be carried by the water flowing in the bedrock. The radionuclides may interact with the rock by mea ns of sorption onto the surface of the fissures nd by diffusion nto •he rock matrix and sorption onto the surfaces f the inner microfissures. The sorption of radionuclides on the rock and tl•e diffusion into it have been studied n laboratory experiments. he trans- port through fissures n the rock has been studied both in the laboratory and in in situ experiments Neretnieks et al., 1982; Abelin et al., 1984]. The aim of this study s to test the capability of two models to predict the transport of Strontium hrough a single issure using data from experiments carried out in the labo•ratory. Velocity variations are modeled by hydrodynamic dispersion in one model and by channeling ispersion n the other model. The influence of the selected model, when the results from these ests are used o predict breakthrough curves or longer migration distances, s also studied. EXPERIMENTAL Two granitic rill cores aken rom the Stripa mine were used n the experiments. ach drill core had a natural issure running parallel to the axis. The dimensions f drill cores were core A, 18.5 cm in height and 10.0 cm in diameter, and core B, 27.0 crn n height and 10.0 cm in diameter. Figure 1 shows he experimental setup. Tracers were inject- ed into the upper channel in core A and into the lower channel in core B. The outlet channel was simultaneously flushed with water to reduce he delay time due to the channel volume. The water residence time in the inlet channel as well Copyright 1985 by the American Geophysical Union. Paper number 5W0275. 0043-1397/85/005W-0275505.00 as n the outlet channel s important ompared ith the resi- dence ime in the fissure proper. The channel used o distrib- ute he racer long he issure nlet had a volume f 1.1 mL and the volume of the outlet channel was about the same. The experimental echnique s described n some detail by Neret- nieks t al. [1982]. The tritiated water, odide, romide, nd lignosulphonate ons were selected as nonsorbing racers. Strontium was used as a sorbing on. The injection flow rates were between .22 and 1.62 mL/min for the nonsorbing tracers. For the tests with sorbing tracer the flow rates were between 0.75 and 1.25 mL/min. The experiments were per- formed t the department f Nuclear Chemistry y Eriksen and coworkers. The porosity of Stripa granite and diffusivity of the iodide tracer and the tritiated water have been determined by Ska- gius nd Neretnieks 1982, 1983]. A piece f granite -10 mm in thickness was fixed between two chambers. One chamber contained a solution with a high concentration of the non- sorbing ion and the other chamber with a very low con- centration of the ion. The diffusivity s determined rom the rate at which the ion is transported hrough the piece of gran- ite. The porosity was determined by weighing dry and wet samples. The sorption of strontium on crushed granite was deter- mined by Skagius t al. [1982]. Different article ize ractions (from 0. D to 5.0 mm) were Used. he particles were equilibrat- ed or more han 1 year with raced olution. he arger arti- cles had an equilibrium ime of about 1 year. From these data surface and volume distribution coefficients nd diffusivity of strontium were determined. The volume distribution coef- ficient and diffusivity of strontium for pieces of granite have also been determined Skagius nd Neretnieks, 983], Pieces of granite 5 mm in thickness were used. CONCEPTS oF THE MODELS The transport of a species carried by a fluid flowing in a fissure n rock is influenced by (1) molecular diffusion n the liquid, (2) variations n fluid velocity n the fissure, 3) velocity variations between channels n the fissure, and (4) chemical or 951  952 MORENO ET AL.' TRACER TaTS IN NATURAL FISSURES .••endp•ece _• utlet hssure flushing J water J e••jper•stalt• J pump n m ' - ..... __•solubon frachonal collector Fig. 1. Experimental setup. physical nteractions with the solid material. Two models are used to describe he transport through a thin fissure. Both models describe he tracer transport as taking place through a parallel-walled fissure. The tracers penetrate the matrix by molecular diffusion and they may be sorbed onto the fissure walls and onto microfissure surfaces within the rock matrix. The transversal dispersion n the plane of the fissure s as- sumed to bc negligible. The tracer movement n the fissure proper is studied considering wo different models: (1) the hydrodynamic dispersion-diffusion model and the (2) channeling dispersion-diffusion odel. The first model assumes hat tracers flow in a parallel- walled fissure, and the spreading of the tracer in the fissure s taken into account by means of hydrodynamic dispersion which s modeled as Fieklan dispersion. n the second model t is assumed hat the fissure consists of parallel unconnected channels n the plane of the fissure, he channel widths having a lognormal distribution. This causes he tracer to bc carried different distances n a given time due to the velocity vari- ations between the channels. The velocity variations are duc to the differences n channel width and/or flow resistance. he hydrodynamic dispersion n each channel s assumed o be negligible; .e., n this case he spreading s entirely due to the variation in flow velocity hrough the different channels. he models are shown graphically n Figure 2. The water residence ime in the inlet channel depends on the flow of tracer injection. The ratio of the flow of flushing water to the flow through the fissure determines he water residence time in the outlet channel. If the overall flow through the fissure s thought to be the sum of the flows following several distinct pathways, each pathway will have a different time delay in the inlet channel and in the outlet channel. The pathway closest o the inlet will have the least time delay in the inlet and the greatest n the outlet. Disper- sion for each pathway in the inlet and outlet channels s as- sumed to be negligible. Continuous flushing of the outlet channel makes the residence time there less than in the inlet channel. The flow through he inlet channel, issure, nd outlet channel s divided into various pathways as shown n Figure 3. Each pathway has a different ime delay. GOVERNING EQUATIONS H ydrodynamic ispersion-Diffusion odel The model considers he transport of contaminants in a fluid that flows through a thin fracture in a water-saturated , ROCK Fissure I Water flow Same velocity and hydrodynamic dispersion in all channels Fig. 2a. Hydrodynamic dispersion-diffusion odel. porous rock. The tracers may be sorbed on the fracture sur- face by an instantaneous eversible eaction. The tracers may also penetrate he rock matrix by molecular diffusion and may be sorbcd onto the microfracture surfaces within the rock matrix. The following processes ill be considered: 1) advec- tive transport along the fracture, 2) longitudinal mechanical dispersion n the fracture, 3) sorption onto the surface of the fracture, 4) molecular diffusion rom the fracture nto the rock matrix, and (5) sorption within the rock matrix. By assuming a linear isotherm for the surface sorption, the differential equation for the transport of a tracer n the fissure may be written n the ollowing way: 3C•. •. 2C•. •. ?C•. D CpJ •t Ro •x 4 Ro c•x 6 R. c•z = = O (1) where Ro s the surface etardation coefficient efined as 2 Ro=I +•Ko (2) Other coefficients are defined in the notation. The differential equation or the porous matrix is c•Cp De c•ZCP 0 (3) c•t Kdp c•z Kd is the bulk distribution coefficient; t is based on the mass of microfissured solid and includes the nuclide which is on the solid as well as in the water in the microfissures. a is related to the distribution coefficient based on the mass of the solid proper, Ka', by K•pp = ep + K•'pp (4) Kdpp s usually he entity determined n sorption measure- ments. he difference etween a'pp nd K•pp s negligible or sorbing tracers on low-porosity materials such as crystalline rock; K•'pp >> p or such ystems. The system of (1) and (3) with the initial and boundary conditions existing n the tracer tests has an analytical solu- tion [Tang et al., 1981-1. he initial and boundary conditions • ..... _•_ /"'ROCK •// water tlOW - Different velocities in different pathways. No dispersion. FiS. 2•. Channelins dis•rsion-di•usion m•el.  MORENO ET AL.' TRACER TESTS N NATURAL FISSURES 953 are zero initial concentration, constant concentration at the inlet during the tracer injection, and a matrix and fissure of very large extension. The tracer concentration n the outlet may be written as C(t)/C = (2/x/•) exp Pe/2) exp _•2 _ pe2/16•2) ( (Pe o/8A•2) rfc t- (Pe o/4•2)} /2' • (5) where l = (Pe to/4t) M (6) Pe = U x/D•. (7) t o = Ro x/Uf = Rot• (8) A = 6Rff2(DeKdpp) M (9) Channeling ispersion-Diffusion odel In this model the dispersion hat occurs n the direction of the flow is accounted or by means of channeling dispersion. The velocity differences n the channels will carry a tracer different distances ver a given ime. The transport of the tracers akes place through a fracture in which parallel channels with different widths exist. This is shown in Figure 2b. It is assumed hat the fissure aperture widths have a lognormal distribution and the interconnection between he different channels s negligible. he hydrodynamic dispersion n each single channel s also assumed o be negligi- ble compared o the effects of channeling. The model includes the following mechanisms: 1) advective ransport along the fissure, 2) channeling ispersion, 3) sorption onto the surface of the channels, 4) diffusion into the rock matrix, and (5) sorption within the rock matrix. For a tracer flowing through a fissure with negligible ongi- tudinal dispersion, he equation for the concentration n the fissure is c3Cf f 3Cf D 3Ct, = 0 (10) 3t R c3x 6 R c3z =o The equation or the diffusion nto the rock matrix is given as before by (3). The solution for (10) and (3) is found in the literature [Carslaw and Jaeger, 1959]' where Bt,• (11) _ erfc 6(t - -•)l/2• o t o = Rat w (12) B = (DeKctpt,) /2 (13) If, in each pathway separate channels exist with different fissure widths, 6, the fluid will have different velocities in the different channels when flowing through the fissure. n this case Ra will be different or the different channels. he con- stant entity is Ka, the surface distribution coefficient. f the breakthrough curve for each channel n the actual pathway is given as Cf(6, t) then he concentration f the mixed effluent from all the channels n the pathway is [Neretnieks et al., 1982]: © (t) f(6)Q(6)C i6, ) a6 - (14) Cø 6)Q(6) 6 Outlet I I I I I I Injection Different locations of the tracer due to delay in injection channel Flushing Fig. 3. Residence ime distributions n inlet and outlet channels are accounted for in the model. In a parallel-walled hannel of width 6, the flow rate for laminar flow is proportional to the fissure width cubed. Snow [1970], studying he fissure requencies or consolidated ock, found that the fissure widths have a lognormal distribution. The density unction has the form 1 1 ([ln(6/#)]2.) (6) o.(2•r)1/2 exp (15) a 2 FIT OF THE EXPERIMENTAL DATA Breakthrough Curve or the Effluent To account for the influence of the finite volumes of the inlet and outlet channels, ime delays must be considered. n each pathway the time delay is determined by the distance o the tracer inlet. A dimensionless istance between he position of the respective pathway and the inlet is defined: W (see Figure 3). The time delay for a pathway may be written as t n = tn•(• , q) + tn2(•, q, g) (16) where q is the water flow rate through the fissure, and g is the ratio of the flushing low to the flow q. The breakthrough curve at the outlet or a given pathway s C = C'(t - tD)= C(W, t) (17) and the concentration of the mixed effluent rom all the path- ways s 1 (t) = C(W, t) dW (i 8) Determination of the Parameters The concentration of the effluent rom all the pathways for the hydrodynamic dispersion-diffusion model may be written as C(t) -f(Pe, t,•, Ka, A, t) (19) Co For the channeling dispersion-diffusion model the con- centration of the effluent becomes C(t) -f (a, œw, ,,, B, t) (20) Co The determination of the parameters was done by means of a nonlinear least squares fitting. The runs with nonsorbing tracers were used for the determination of the hydraulic  954 MORENO ET AL.' TRACER TESTS IN NATURAL FISSURES TABLE 1. Experiments with Nonsorbing Substances Fitted With the Dispersion-Diffusion Model, Core A Run Tracer Water Standard Peclet Residence Fissure Deviation Number Time Width of Fit Pe t,•, rain 6, mm s/Co A1 NaLS A2 NaLS A3 NaLS A4 NaLS A5 NaLS A6 NaLS A7 NaLS A8 NaLS A9 NaLS A10 NaLS All 3-H A12 3-H A13 3-H A14 I A15 Br 15.7 1.22 0.13 0.01 11.2 1.79 0.14 0.01 14.2 2.68 0.14 0.04 9.5 3.52 0.14 0.03 91.5 5.26 0.14 0.02 36.8 10.0 0.15 0.02 59.0 5.00 0.14 0.01 26.2 5.44 0.15 0.03 14.4 10.1 0.15 0.02 15.5 5.07 0.14 0.01 109.0 2.99 0.16 0.03 43.0 5.94 0.16 0.03 85.2 11.9 0.17 0.04 15.0 2.42 0.13 0.02 29.6 2.61 0.14 0.01 D e = 0.1 x 10 •2 m2/s. properties. These properties Pe and tw for the hydrodynamic dispersion odel and a and •w or the channeling odel) re used n the runs with the sorbing racer to determine he other parameters. When the water residence ime tw (or equivalent fissure width) is known from the nonsorbing racer runs, the surface quilibrium oefficient, a, and the product DeKap are obtained from the sorbing racer fits. The modelling considers he existence of various pathways with different time delays. These time delays are calculated considering hat the inlets for the tracer and the flushing water are on the same side of the respective channels and that the outlet is on the opposite side of the inlet; these ocations are shown n Figure 3. First, the experimental data was fitted using the hy- drodynamic dispersion-diffusion model. For the tracer tests with nonsorbing ubstances, he determination f the parame- ter which takes into account the interaction with the rock matrix, the A parameter, cannot be determined with any accu- racy. The reason for this is the short tracer residence ime, which results in a small interaction with the matrix for a nonsorbing tracer. The water residence imes were in the range 2-10 min. In the runs with nonsorbing tracer the A parameter was calculated assuming a value of De=0.1 'FABLE 2. Experiments with Nonsorbing Substances itted With the Dispersion-Diffusion Model, Core B Run Tracer Water Standard Peclet Residence Fissure Deviation Number Time Width of Fit Pe t,•, min 6, mm s/Co B1 NaLS B2 NaLS B3 NaLS B4 NaLS B5 NaLS B6 NaLS B7 3-H B8 3-H B9 3-H B10 I Bll Br 18.6 1.94 0.14 0.01 9.9 2.24 0.12 0.02 14.9 3.67 0.12 0.02 80.2 4.56 0.12 0.01 14.4 7.14 0.13 0.01 13.2 15.3 0.15 0.02 15.8 4.55 0.15 0.01 48.2 8.97 0.16 0.02 40.0 18.1 0.18 0.03 9.5 4.07 0.13 0.03 9.6 4.30 0.14 0.03 D e -' 0.1 x 10 •2 m2/s. Fig. 4. o.o 5.0 o.o TIME, MIN Curves fitted with the hydrodynamic dispersion model for nonsorbing racers runs A1, A2, and A3). x 10-12 m2/s. This value of the effective iffusivity as a very small impact on the breakthrough curve compared to when De- 0; i.e., the diffusion into the matrix for a nonsorbing tracer with short residence ime is negligible for a value of De- 0.1 x 10 12 m2/s and less. Diffusion values or iodide and tritiated water were determined by Skagius nd Neretnieks [1982] for pieces of granite. These values are in the range 0.07-0.18 x 10 12 m2/s. The effective iffusion oefficient or the lignosulfonate on is assumed o be of the same order of magnitude or less han the iodide and tritiated water because this molecule s very large. A large molecule would have low access o the micropores of the rock matrix. The surface sorp- tion coefficient s Ka = 0 for a nonsorbing racer. The other parameters Pe and tw) are determined by means of a data fitting. From these values the dispersion coefficient and the average issure width are calculated. The fissure width is directly btained rom he measured low rate and he resi- dence ime tw. The results are shown n Tables 1 and 2. Some fitted curves are shown in Figures 4 and 5. For core A the average Peclet number was about 20 and the fissure width was 0.14 mm; for core B these values were 15.0 and 0.13 mm, respectively. For the runs with sorbing racer the values of the Peclet number and fissure width determined in the runs with the nonsorbing tracers are used. The tracer residence ime to-- Ratw and the parameter A which takes nto account he inter- action with the rock matrix are determined by a data fitting. c•. + ß ß ..x. x o- x x .-x .............. + 1•,, 0.0 lO.O 20.0 TIME, MIN Fig. 5. Curves fitted with the hydrodynamic dispersion model for nonsorbing racers runs B 1, B2, and B3).  MOPENO T AL.' TRACER •STS N' NATURAL FIssug• 955 TABLE 3. Experiments With the Sorbing Substance Strontium TABLE 4. Experiments with Nonsorbing Substances itted With Fitted With the Hydrodynamic ispersion-Diffusion odel the Channeling ispersion Model, Core A Run* Tracer Surface Standard Standard Water Mean Standard Residence Sorption Deviation Deviation Residence Fissure Deviation Time A Coefficient, DeKdpr, of Fit in Lognormal Time Width of Fit t o, min Parameter K a x l0 s m 1012 m 2 s/C Run Distribution t• min $, mm s/C A16 12.7 33.5 27 101 0.05 A17 14.3 65.1 31 34 0.07 A18 8.4 44.2 27 58 0.06 B12 13.2 70.5 16 10 0.02 B13 11.7 43.7 13 20 0.02 B14 6.6 42.0 12 19 0.02 *The Peclet numbers are 20 for core A and 15 for core B. $D•Kdpr etermined n the laboratory s 3.5 x 10 12 m2/s or crushed ranite particles nd 2.4 x 10 x2 m2/s or sawn pieces f granite. The inlet concentration s obtained directly from the experi- mental data (concentration f the solution, njection low and flushing low). The results are shown n Table 3. The break- through curve of a run with strontium s shown n Figure 6. The surface orption oefficient as about 28.3 x 10 s m for core A and about 13.7 x 10 -s m for core B. The value determined in the laboratory [Skagius et al., 1982] is 6.6 x 10 • m. From the value of the A parameter t is only possible o determine he product DeKaPv. n core A it was about 6.4 x 10 • m2/s, while t was about 1.6 x 10 • m2/s in core B. The values determined n the laboratory were 2.4 x 10 •2 m2/s for sawn pieces of granite and 3.5 x 10 •2 m2/s when hese alues were determined sing rushed ranite particles 5 mm in diameter. The sawn and crushed pieces had no coating material. The channeling ispersion-diffusion odel was used only to evaluate he experiments n core A both for the nonsorbing substance NaLS) and sorbing substance strontium). n the runs with the nonsorbing racer NaLS the B parameter was calculated sing value of 0.1 x 10-x2 m2/s or the effective diffusivity of the lignosulphonate on. The other parameters were calculated y means f a fitting process a and •). The results for the nonsorbing substance are shown in Table 4. The mean fissure width was 0.14 mm. This same value was obtained when the hydrodynamic dispersion model was used to fit these data. The mean standard deviation for the loga- rithm of the fissure width was a - 0.155 or these experiments. To compare he values f a and Pe determined rom he it A1 0.176 1.27 0.133 0.01 A2 0.206 1.87 0.151 0.01 A3 0.186 2.79 0.149 0.03 A4 0.220 3.71 0.150 0.03 A5 0.055 5.32 0.130 0.02 A6 0.119 10.2 0.148 0.02 A7 0.096 5.08 0.124 0.02 A8 0.138 5.57 0.151 0.03 A9 0.180 10.5 0.152 0.02 A10 0.181 5.30 0.130 0.01 D• = 0.1 x 10-x2 m2/s; NaLS racer. a theoretical elationship s determined. A breakthrough curve is created by the channeling model for different values of a. The curves so created are then used as if they were experi- mental results. The Peclet number is determined from the first and second moments of the breakthrough curve as ELev- enspiel, 1972] 2/Pe = ttt2/t 2 (21) where or a step unction njection c = o (oo) c(t) (•) at (22) at = 2 t [C(c•)- (t)] t, C(c•) dt (23) It is possible o find an analytical expression etween Pe and a 2 [Neretnieks, 983]' 2 • = exp 4a 2) -- 1 (24) Pe This can be used o give he relation between e and a 2 and, of course, gives the same results as the method described above. n the curve itting method, however, other information can also be obtained such as higher moments. The relation so obtained is shown by the full line in Figure 7. The values 0.0 50.0 100.0 TIME, MIN Fig. 6. Curves itted with the hydrodynamic ispersion model solid line) and channeling model (dashed ine) for strontium run A16). o.o' g.o so.o A.o ,oo.o PECLET NUMBER Fig. 7. Relationship etween Peclet number and standard deviation of the logarithm of the fissure width.
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