I40 RHO-BW-CR-131 P Analysis of Two-Well Tracer I. Tests With a Pulse Input Lynn W. Gelhar, Consultant Prepared for Rockwell Hanford Operations, a Prime Contractor to the United States Department of Energy Under Contract DE-AC06-77RL01030 K> ~~~91 9210150247 920914 i PDR WASTE PDR I441 Rockwell International Rockwell Hanford Operations Energy Systems Group Richland, WA 99352 A a. I . M. -I
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Analysis of Two-Well Tracer Tests with a Pulse Input.
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I40RHO-BW-CR-131 P
Analysis of Two-Well Tracer
I. Tests With a Pulse Input
Lynn W. Gelhar, Consultant
Prepared for Rockwell Hanford Operations,a Prime Contractor to theUnited States Department of EnergyUnder Contract DE-AC06-77RL01030
K> ~~~91
9210150247 920914i PDR WASTE PDRI441
Rockwell InternationalRockwell Hanford OperationsEnergy Systems GroupRichland, WA 99352
A a. I . M. - I
Rockwell InternationalRockwell Hanford Operations
Energy Systems GroupRichland, WA 99352
DISCLAIMER
This report was pnrepamd as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied. Or assumes env legal liability or respones.bilitv for the accuracy, completeness. or usefulness of any information. apparatus, product. orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process. of service by trade name, trademark.manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The viewsano opnions of authors expressed herein do not necessarily state or reflect those of the UnitedStates Government or any agency thereof.
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.sANALYSIS OF TWO-WELL TRACER TESTS
WITH A PULSE INPUT
Lynn W. Gelhar, ConsultantCambridge, Massachusetts
April 1982
Prepared for the United StatesDepartment of Energy UnderContract DE-AC06-77RL01030
Rockwell IntemationalMOWe NaMord OperaUbta
IANwY Sysmiw am"uP.O. 6ox go0
ftdid. Wilson 30932
RHO-BW-CR-131 P
SUMMARY
Dispersion of a conservative solute, which Is Introduced as a pulse in
the recharge well of a two-well flow system, is analyzed using the general
theory for longitudinal dispersion in nonuniform flow along streamlines.
Results for the concentration variation at the pumping well are developed
using numerical integration, and are presented in the form of dimensionless
type-curves, which can be used to design and analyze tracer tests.
Application of the results is Illustrated by analyzing the preliminary
tracer test run at boreholes DC-718 on the Hanford Site by Science
Applications, Inc., a subcontractor to Rockwell Hanford Operations, in
The objective of this analysis is to describe the tracer concentrationwhich evolves in the pumping well of a two-well (pumping-recharge) flowsystem (Fig. 1) when an instantaneous pulse (slug) of conservative tracer isintroduced in the recharge well. The streamline pattern for steady flow ina homogeneous confined aquifer is used in conjunction with the generaltheoretical results of Gelhar and Collins (1971) for longitudinal dispersionin nonuniform flows. With a pulse input their general result for theconcentration, c, is:
c(st) U u(s0 ).iri exp -n(1)
where
s = distance along streamline
t time
a longitudinal dispersivity
n T(S) - t
s'T(s) ' f ds/u(s), traveltime to s
SO
w(t) - f ds/Cu(s)J 2
s(t) = mean location of the pulse at time, t
u(s) - seepage velocity
m a mass of tracer per net area of aquifer injected at s zs- at time, t =O.
1
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PUMPING RECHARGEWELL WELL RCP8305-30
FIGURE 1.with Q/Qr
Streamline Pattern for= 2/3.
Two-Well Flow System
Equation 1 is applied along each streamline identified by the value ofthe stream function, 4'; therefore the velocity, u, depends on 1, and as aresult, T(s, 4) and w(t, p). These flow integrals can be evaluated eitheranalytically or graphically as described in the next section. The coeffi-cient m/u(so) in Equation 1 is evaluated by noting that at the recharge well:
u(so) - Qr/( 2wrwnH)
Qr recharge rate
rw = well radius
n = effective porosity
H = aquifer thickness
m = M/2wrwnH
M = mass of tracer injected
m/u(s 0) = M/Qr
2
RHO-BW-CR-131 P
The concentration in the pumping well is found by calculating the flow-weighted concentration as the following. integral:
c 2H fQ/2H(M)(4w=)e1/2xp [-( - t) /4=]d (2)
where B ' Qr/Q. In this integral, the flow integrals, T and w, depend onthe stream function, 4.
2.0 EVALUATION OF FLOW INTEGRALS AND WELL CONCENTRATION
For the case of equal recharge and discharge rates (B - 1), theintegrals for T and w can be evaluated analytically from the velocityvariation along a given streamline. The details of this analysis are givenin Appendix A; the results for T and w are given by Equations A3 and A4expressed in dimensionless form as:
a(¢) = , b(s,*)='(- 2 (3nHL L
where tw c traveltime to the pumping well. The well concentration isevaluated using these dimensionless forms and the variable of integrationV - */(BQ/2H). Then, for the upper half plane in Figure 1, the flow fromthe recharge well is represented by the range 0 < V < 1. Furthermore, theconcentration will always be zero for the streamTines with 0 1 becauselateral dispersion is neglected. Then, using Equation 3 in Equation 2,
M 1 Q.exp -(a-T)2/4eb] dAcw, ZI 2 172nL(4ireb)
nHL2 1 exnt-fa T)2/4d,1AC Z -wr CW C I e t-(aT b do
CO ~(4rc b)
T Qt C Er (4)nHL
K>
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The integral in Equation 4 was evaluated numerically using a and b fromAppendix A; a listing of the computer program is included in Appendix C.
For the case of unequal flow, i and w were determined graphically froma flow net constructed for the case 0 - 2/3 as described in Appendix B.
The limiting result for no dispersion in Equation 4 is found by notingthat, as E * 0,
expf-(a-T 2/4cb] * 6(a-T
(4wcb) /2
where 6(x) = Dirac delta function.a (iT), Equation 4 becomes:
Changing the variable of integration to
A f0
a(o)6(a-T) Ka da C TlacT' T >a(o)
0, T <a(o)
(5)
The function a(Q) is simply the dimensionless traveltime to the pumping wellalong a given streamline, $; the general form of this function is shown inFigure 2. The non-zero portion of Equation 5 can be evaluated by taking:
A (dao -1
da (;and c(T)assigned
is then determined by taking T a(V ) and da/d*q1$, where t1value of V.
is an
3.0 RESULTS
Numerical evaluation of the concentration in the pumping well as givenby Equation 4 was carried out using the computer program listed inAppendix C. Several runs were made to test the effects of model parametersand approximations. The results are listed in tabular form in Appendix 0.
4
RHO-BW-CR-131 P
The integral in Equation 4 was evaluated numerically using a and b fromAppendix A; a listing of the computer program is included in Appendix C.
For the case of unequal flow, T and w were determined graphically froma flow net constructed for the case 0 - 2/3 as described in Appendix B.
The limiting result for no dispersion in Equation 4 is found by notingthat, as c * 0,
exp[-(a-T)2/4EbI * 6(a-T)
(4wffcb) 112
where S(x) = Dirac delta function.a (Vt), Equation 4 becomes:
Changing the variable of integration to
cu fa(o)
&(a-T) Si da - dl a=T, T >a(o)
0, T <a(o)
(5)
The function a(¢) is simply the dimensionless traveltime to the pumping wellalong a given streamline, i; the general form of this function is shown inFigure 2. The non-zero portion of Equation 5 can be evaluated by taking:
A -Oa di/
and c(T)assigned
is then determined by taking T = a (t ) and da/d|Iq 1, where $t i is anvalue of V.
3.0 RESULTS
Numerical evaluation of the concentration in the pumping well as givenby Equation 4 was carried out using the computer program listed inAppendix C. Several runs were made to test the effects of model parametersand approximations. The results are listed in tabular form in Appendix D.
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<a3 0.5
0I0 1 2 3 4 5
A
(WI RCPS305-31
FIGURE 2. Dimensionless Traveltime Function.
Figure 3 shows a comparison of the graphically based results usingm I (Appendix B) and the exact result using the equations from Appendix A.The excellent agreement demonstrates the adequacy of the graphicapproximation with m - 1 In Equation 52; that value was used in allsubsequent calculations. Figure 4 illustrates the effect of using theapproximate form of Equation 62; some difference is observed at lowerconcentrations for the rising limb of the curve with the large value ofc * 0.2. For smaller c, the differences are generally smaller, as shown inFigure 3. The effects the increment, AV, used to approximate the integralin Equation 4 are illustrated in Figure 5. When E = 0.01, the largerAt = 0.05 produces oscillations in the tail of the curve, but these areeliminated when At * 0.01; for E = 0.2 the results are nonoscillatory whenAV = 0.05. Generally, the oscillations are eliminated if AV < c.
The overall results are summarized in Figures 6 and 7; the unequal flowcase (Fig. 7 with B = 2/3) corresponds to the Science Applications, Inc.tracer test of December 1979. These results show that the dispersionparameter, c a v/L, affects the rising part of the curve and the peak butnot the tail. This point is further demonstrated by the behavior of thenondispersive solution using Equations 5 and AS As shown in Figure 6. Thisshows that all of the results approach the nondispersive analytical resultfor large time, and further demonstrates the adequacy of the numericalprocedure. Generally, the breakthrough curves are characterized by a steeprising limb and an elongated tail, as shown in the linear plots of Figure 8.'The log-log plots (Fig. 6 and 7) are convenient for estimating thedispersivity and effective porosity from tracer test data, as illustrated inthe following section.
5
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. I
it I
0.o0 LI0.1 0.5 1 5
T -pHI.1
10
RCPS30S-32
FIGURE 3. Comparison of Exact and Graphic Flow Net Calculations.
6
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I
go
0.5
<a 0.1
0.06
0.01 L_0.1 0.5 I 5
T
10
RCP8305-33
FIGURE 4. Comparison of Exact Calculation With ApproximationUsing Equation B2.
7
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I
0.5
<0 0.1
0.05
0.01 It0.1 0.6 1 5
T10
RCPa305-34
FIGURE 5. Effect of Integration Increment.
8
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1-
0.5 _
0.1
0.1 0.5 1 60:
T at-a
la
RCPS305-3S
FIGURE 6. Type-Curves for Two-Well Pulse Input Test With Equal Flow.
9
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-_
0.5
<U
0.05 _
0.01 _0.1 0.5 1 6
. (tT -
10
RCPS305.36
FIGURE 7. Type-Curve for Two-Well Pulse Input When the RechargeRate is Two-Thirds of the Discharge Rate, Q.
kJ
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- EQUUAL FLOW|_| Lu *0.01. 0.02. 0.05. I
0.01
t 0.5
0.02
0.05
0.2
0 6 lo
T *a RCPS305-37
FIGURE 8. Pulse Input Type-Curves Plotted With Linear Scales.
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4.0 APPLICATION METHODOLOGY
The procedure for interpretation of tracer test data using the resultsof the above analysis is illustrated by analyzing the test conducted byScience Applications, Inc. at DC-7/8 in December 1979. Since some of theconditions of that test were not fully defined, the results of this interpre-tation are considered to be preliminary; this example is present primarilyto illustrate the procedures which can be used to analyze such tests.
The Science Applications, Inc. tests were reanalyzed using the type-curves in Figure 7. The flow rates for the test were estimated using theaverage rates implied for the period 15:13-23:00 (see Appendix E, Table E-4);Qr = 2.31 gal/min (injection rate) and Q - 3.42 gal/min (pumping rate) orQr/Q -B a 2/3. Based on these rates and estimates of the volume in theborehole flow conduits and connecting plumbing, the following traveltimeswithin the tubing to and from the test horizon were estimated:
* Time down in Injection well = 153 min.
* Time up in pumping well * 258 min.
These times were subtracted from the observed times to give the actual elapsedtime since the tracer entered the formation. Also, the elapsed time wascorrected to correspond to a constant pumping rate of 3.42 gal/min, based onthe actual metered volume in Table E-6. After these time corrections weremade, the actual data points in Figure E-2 were plotted with the estimatedbackground of 20 counts subtracted; the corrected data are shown inFigure 9, along with two of the dimensionless type-curves in Figure 7 forB = 2/3. Overlaying the data on the type-curve, we find a reasonable fitfor E in the range 0.02 to 0.05; using e = 0.035 = a/L and L = 56 ft, theindicated longitudinal dispersivity is a 0 9.035(56) = 1.96 ft. Matchingthe time scales in Figure 9, I - 1 * Qt/nHL, t(hours) -.1.18 hr yields theeffective thickness nH = Qt/L - 0.0105 ft.
The data in Figure 9 seem to show some systematic departures from thetheoretical type-curve. This could be a reflection of experimental ambigui-ties such as the following:
1. The background concentration is not clearly determined, and smallchanges in this level could drastically alter the lowconcentration parts of the curve.
2. Unobserved flow rate variations during the period that the tracerwas passing the sensor would distort the shape of the curve.
3. Uncertainty about the volume in the connecting conduits couldintroduce errors in the traveltime correction and alter the shapeof the curves.
12
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1.000
500
100
60
J1
10
5
20.1 0.6 1 5
TIME (hrl10
RCP8305.38
FIGURE 9. Type-Curve Matching.
13
RHO-BW-CR-131 P
*The differences in Figure 9 could also indicate that the tested zone doesnot behave as a homogeneous, constant-thickness aquifer. If other sourcesof errors were eliminated, departures on the tail of the curve would bediagnostic of that possibility because that part of the type-curves isdetermined solely by convection; i.e., the traveltime distribution.
Calculations for the Science Applications, Inc. test were also madeusing the approximate method developed in Appendix E. From Figure 9 thepeak time t = 1.5 hr and the time to rise from one-half of the peak ist = C-45 hr. Then using Equation E35 with F = 0.202, G 0.0488 (a = 2/3):
a r 1 F (At) = 0.0271r 41nZ G tZ
p
a = 1.52 ft
and from Equation E22 with Q = 3.42 gal/min:
QtnH = i = 0.0103 ft.
2,1 F
These results show good agreement with those from the type-curve approachand indicate that the approximate method in Appendix E is reliable. Ofcourse, the type-curve method has the advantage, in that it uses thecomplete breakthrough curve.
The type-curves of Figure 6 and 7 can also be used to design tracertests; using estimates of c a a/L, the actual concentration level that willresult from a given mass of tracer M can be determined in terms of theeffective thickness nH and the well spacing L; i.e., cw a Mt/nHL2.
5.0 COMMENTS AND RECOMMENDATIONS
These results demonstrate the feasibility of the two-well tracer testwith a pulse input as a method of determining effective porosity anddispersivity. This type of test has the advantage that the shape of thebreakthrough curve is very sensitive to the dispersivity. This is incontrast to the more frequently used step input (Webster et al., 1970; Groveand Beetem, 1971; Robson, 1974; Mercer and Gonzalez, 1981) in whichdispersion affects the shape of the curve only in the initial lowconcentration portion of the curve.
14
RHO-BW-CR-131 P
The type-curves developed here provide a simple method of designing andanalyzing two-well pulse input tracer tests. I
The method of analysis used here presumes that a/L is relatively small;results in Gelhar and Collins (1971) indicate that the method should bereasonably accurate for v/L < 0.1. If the method is to be used for largervalues of a/L, some comparative testing with numerical solutions issuggested. However, it should be recognized that, under those conditions(large a/L), other factors such as displacement-dependent dispersivity andnon-Ficklan effect (Gelhar et al., 1979) may complicate the interpretation.Also, transverse dispersion is neglected in this analysis; this assumptionis reasonable for small a/L because then the dispersion effect occursprimarily along the more direct streamlines for which the fronts will benearly perpendicular to the streamlines. For larger a/L, the dispersioneffect along a wider range of streamlines becomes important; numericaltesting would also be required in this case. When a/L is large, a finite-difference or finite-element solution should be routine because a relativelycoarse grid could be used.
The type-curves for the pulse input can also be used to treat otherinputs by convolution. In particular, this would apply to recirculatingtests in which the pulse is routed through the aquifer several times. Thisaspect is important in the Hanford tests because analysis of the secondarypeaks would provide a check on the borehole traveltime. Some preliminarywork has been done on numerical convolution of the pulse input results. Itis recommended that this be developed for analysis of the Hanford data.
6.0 REFERENCES
Gelhar, L. W. and M. A. Collins, 1971, "General Analysis of LongitudinalDispersion in Nonuniform Flow," Water Resources Res., 7 (6),pp. 1511-1521.
Gelhar, L. W., A. L. Gutjahr, and R. L. Naff, 1979, "Stochastic Analysis ofMacrodispersion in a Stratified Aquifer," Water Resources Res., 15 (6),pp. 1387-1397.
Grove, D. B. and W. A. Beetem, 1971, "Porosity and Dispersion ConstantCalculations for a Fractured Carbonate Aquifer Using the Two-WellTracer Method," Water Resources Res., 7 (1), pp. 128-134.
Mercer, J. W. and D. Gonzalez, 1981, Geohydrology of the Proposed WasteIsolation Pilot Plant in Southeastern New Mexico, Nei- Mexico GeologicalSociety Special Publication 10, pp. 123-131.
Robson, S. G., 1974, Feasibility of Water-Quality Modelina Illustrated byApplication at Barstow? California, Water Resources Invest. Rep. 46-73,U.S. Geol. Survey, Menlo Park, California, 66 pp.
Webster, D. S., J. F. Proctor, and I. W. Marine, 1970, Two-Well Tracer Testin Fractured Crystalline Rock, U.S. Geol. Survey Water Supply Paper1544-1, 22 pp.
15
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APPENDIX A
T AND X FOR EQUAL FLOW CSE*. t. '....
A-1
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APPENDIX A
T AND w FOR THE EQUAL FLOW CASE
When Q u Q it is easily shown that the streamlines are circular arcs..: that case the few is conveniently described in the polar coordinate systemshown in Figure A-1. The piezometric head, h, for steady flow in this systemis:
In
h = Tlk ln(rl/r2)2
Tr a transmissivlty
r21 (x + L)2 + y2, x = -R siny
r2 = (x - L)2 + y2, y - R cosy - 8
and using the Darcy equation, seepage velocity along the streamline is:
. Tr ohy F-NH
i r21;T Sr2
r2 /
After extensive algebraic manipulation, this reduces to:
V C 2QR.sin3t
Y irnHL 2 (cosy - coso)
Using this velocity in the traveltime integral:
(Al)
so
dsuTS)
y=f-
Rdyvy
a i n HL (siny + sin - (y + *)coso)Q 2i 3 (A2)
K>;/
A-2
Ck C (I.
II
A,I
I
f
iI ,
I ,
I .
-.1-1It
I
II
.4t*
I..
'1%
L.
A'p- I Ax0
LA
O-A
'A)
RCP8305-39
FIGURE A-1. Polar Coordinate System for the Equal Flow Case.
RHO-BW-CR-131 P
When y - *, Equation A2 gives the traveltime to the pumping well or:
a(A) . = -A 3I- (sinf - ~cos¢)
AM"VOWUgS % >;:fir - -. - .
(A3)
Equation A3 gives the dimensionless traveltimeof ~; this result is used for a in Equation 4.integral is evaluated from Equation Al as:
between the wells as a functionSimilarly, the w flow
Here y indicates the position of the pulse along-the streamline correspondingto * vrl. At a given time the position j is found by solving Equation A2for y = ? when T = t; this is done iteratively using the program listed inAppendix C. When j>, the pulse for a given streamline has moved into thepumpinj well; in this case w (or b in Equation 4) was calculated from Equation A4using y = 0.
If dispersion is neglected, the concentration is found from Equation 5using Equation A3 to find dq/dala=T,
da = ,2 s 2A 3A 2 A ^ 3 4 A
-^ a recc cotw4* + 2w Vcsc irf* cot ir4 + V *csc #4d ip -
(AS)
The. concentration cis then found by assigning values of T between 0 and 1,calculating T = a from Equation A3 and dx/da from Equation AS.
A$W A." " -- -- *.- -
A-4
RHO-BW-CR-131 P
APPENDIX 8
GRAPHIC EVALUATION OF T AND X
K>~~~~~~~ .-... ......
F ----.. --;£,e
B-1
RHO-BW-CR-131 P
APPENDIX B
GRAPHIC EVALUATION OF T AND w-. t v'- -;'}w~k2b'
t .- . w--;* -:
The flow integrals t and w can be determined from a graphic constructionof the streamline pattern. This approach is necessary in the unequal flowcase where the analytical description of the flow field is very complicated.The streamline pattern for Qr/Q a = 2/3 (see Fig. 1) was constructed bystandard superposition of the ray streamlines of the appropriate source andsink strength. If each streamtube has a flow, Qo, and a width, w(s), as afunction of the distance along the centerline of the streamtube, s, the velocityis:
u(s) = Q/(nHw(s))
and then the flow integrals are expressed as:
t mfI ds n H Iod
w ds (nH2 f 2ds
These integrals were approximated by measuring the width of the streamtubesat intervals, As, along each of the streamtubes and summing the appropriatequantities (wAs or w2as). The integrals were evaluated for each streamtubefor intervals in the 4 of 0.1 at the pumping and well normalized as inEquation 3. These data for a (f) and b (I) at the pumping well were thenfit to polynomials of the form:
4n2n
ln b E bn 4, (B1)nro
_ , _ _. _.... ... ~ . -.
B-2
RHO-BW-CR-131 P
These expressions were then used in the integrations of Equation 4 to findthe concentration in the pumping well. Note that Equation B1 gives only thevalue of b at the pumping well bw. In general, b will increase with timeas the pulse approaches the weil along a given streamline; this behavior wasrepresented in the form:
b/by . (T/a)m, T <a (B2)s1, T >a
where m is a positive exponent to be specified. a and b from Equations 61and B2 were then used in Equation 4 to find c.
In order to evaluate the above graphic procedure, the flow netevaluation was done first for the equal flow case B - 1 and the results werecompared with the exact analytical approach in Appendix A.
8-3
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APPENDIX C
LISTING OF FORTRAN PROGRAM FORNUMERICAL INTEGRATION OF EQUATION 4
C-1
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C
CCCCcCCCCCCC
CCCcC
CCCCC
CC
ICCC
WellDavid GelharDecember 1981
This program calculates the concentration (Cw) oftracer appearing in a well as a function of the timeelapsed since the tracer was pumped down another well.The user is asked to supply several initializing parameters,and the values of 7 (time) desired. First, the all-purposeparameter epsilon (which accounts for dispersion effects) isinput Then, you are asked to input the last valueof psi and the increment of psi to use. (Psihas something to do with what direction the tracer iscoming from.) Using a smaller delta psi can be moreaccurate. and definitely takes longer. (We areapproximating an integral here. so using smaller steps tendsto be more accurate. Good results are obtained in areasonable amount of time with dosi between 0.05 and 0.2(use smaller dpsi for smaller values of epsilon.))The output of the programs a table of values of time (T)and concentration (Cw). can be sent either to a file(for further processing, plotting etc.) or directly tothe line printer.For each value of 1, a set of values for the parameter Ymust be found. You can choose from two methods: 'Exact'(equal flow only!) which analytically determines y (afunction of psi and 1)3 and 'Nice'. in which Y is set to(T/a)**m where m is a user-supplied constant. (usually 1.0)Next, the program asks how to get the magic sets of valuesA and B. used in the concentration integral. For the equalflow case. these call be found directly as a simpleanalytical function of psi. In unequal flow problems,however, these values cannot be determined exactly. so youmust supply instead five coefficients for a polynomialapproximation to the functinons. Now that initialization iscompleted the program will read in values of T from theterminal and calculate and display Cw. When a negative T isinput, the program will write out the results (on theprinter or to a file called Table.dat) and stop. To look atthe A and B values, or to see T and Cw to more significantfigures, you can look at the files A.dat. a. dat, Cw.dat.T.dat, respectively.
C-2
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c Initialize useful variables and open output filesc FILES USED:c A.dat: A valuesc B.dat: B valuesc Cw.dat: calculated Cwsc 1.dat: T values input by userc W.dat: temporary storage of Ws
real lastamdouble precision pidimension a(O:4).bCO:4).iflag(2)open(unit=36afile-'t.dat'.access='seqout') ! store results hereopen(unit-37.file='cw.dat'.access='seqout') ! for tablepi 3.141592653589793innum = -1 - ! counts number of Ts supplied
c Read in user-supplied parameterswriteC5,1)
1 formattt5 'Input epsilon: '.I)read(5.2) eps
2 format(f6.4)c find out what range of psi values to use
write(5.3)3 formattt5.'Input last psi. and delta psi: '.*)
read(5#4) last.dpsi4 format(3(f6.4))c step evenlys starting at stepsize/2
first - dpsi/2number (last-first+dpsi)/dpsi ! how many values of psi
c Output, in table form. can go either to a file or to thec line printer. Here. we find out which is desired, and openc the appropriate device as unit 38.40 writeS.513))13 formattt5*'Where do you want the output to po?'.s/It1O,
c Y can be calculated in two ways: a nice, simple methodc (using a user-specified fudge factor)} or a more complexc exact method (which works only for the equal flow case).c Find out which method is required and set a flag. If thec nice process is to be used. read in the fudge factor now.80 write(5.14)14 format~t5,'Exact or nice values for y?'./.tlO,
1 'CO=exactl=nice): '.*)read(5.6)y4flag ! set iyflag: 0 exact, I niceif((iyflag.lt.0).or.Ciyflag.gt.1)) goto 80 ! bad responseif (iyflag.eq.O)goto 50 ! exact, no fudge factorwrite(5.15)
wr ito (5,8)8 format(t5.'To stop. input a negative value for t')c MAIN LOOP:c Keep calculating until a negative t is input.c storing values of t and cw in data filet70 write(5,9)9 formattt5. 't 8*)
read(5.10) t10 formot(f21. 10})
innum - innum + 1 ! increment counterwrite(36.10)t write t to file
c as soon as a negative t is input, call a routinec to print the results and terminate
if (t.1t.0) call output(eps.dpsi.firstlast.innum.iFlag.a, b,1 moiyflag)
c get y values for this tif Ciyfflg.eq.1) Call Ynice(numberamat)if (igflag.eq.0) Call Yetact(first.dpsi.number.tapi)
c Here we finally call the routine to get thec number we are after.
result - Cwtt.dpsi.number.pi.eps)write(37,10)result write Cw to filewrite(t.11) result and terminal
11 formatttlO.,'Cw -',f9.5)goto 70 ! loop forever until negative t is inputend ! end of main program
C-5
RHO-BW-CR-131 P
Subroutine Aexact(numberfirstdpsi pi ifl)c This routine calculates values for the parameter ac using an exact equation. The As, one for each valuec of psi used. are written to file a.dat for use laterc in the program. The user specifies whether this routinec or the approximate version 'Afit' is to be used. Notec that the exact method works ONLY for equal flow problems.c initialize variables and open output file
double precision piopen(unit-33.file-'a.dat',access-'seqout') ! open data fileifl 0 0 ! set flag indicating exact methodpsi - first ! initialize psi
c loop through all values of psi, calculating an a for eachdo 1 i lonumber ! how many values we needa = pi (1/sintpi*psi)**2)*(1-pi*psi*(cos(pi*psi)/sintpi*psi)))write(33.2)a ! write it to the file
2 formattf2l.10)psi a psi + dpsi !Increment psi
1 continueclose(unit-33) ! close filereturnend
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RHO-BW-CR-131 P
Subroutine 8exact(number.iflifirst.dpsi.pi)c Bexact gets values for b with an exact equation.c Bs are written to the file B.dat for later use.c Flag 'ifl' is set so the output routine knows thatc exact b values were used. Remember. the exactc method can be used only for equal flow.
double precision phib.piifl - 0 ! set flag: exact valuesopen (unit-34.file='b.dat'.access-'seqout') ! open b.dat
c loop: get a b; write it outdo 1 i - I.numberphi = pi*(first.(x-1)*dpsi) ! phi = psi*pib = pi**2*(phi-3*cos(phi)*sin(phi)+2*phi*(c(stsphi))*+r2)I /(2*(sin(phi))**'3)write (34.2)b output b to file
2 format (f2l. 10)1. continue
close(unit-34) ! close output file b.datreturnend
C'7
RHO-8W-CR-131 P
Subroutine Afit(numberofirstdpsi.ifl.a)c This routine opens the file a.dat and calls.c the routine 'Fit'. which does a polynomial curvec fit using user-supplied coefficients. The fivec coefficients are passed back up to the main programc in the array 'a', and flag ifl is set to indicatec the use of approximate values.
dimension aCO:4) ! coefficient arrayopen (unit-32,file-'a.dat'.access-'seqout') ! open output fileifl = I ! flag approximate a valuescall Fit(numbertfirstodpsi~a) ! call routine to generate Asclosetunit-32) !clean upreturnend
C-8
RHO-BW-CR-131 P
Subroutine Bfit(number.first,dpsi.iflb)c This routine is identical to Afit. but the file b.datc is opened instead of a.dat. Coefficients are returned inc the arra.j'b'.
dimension b(O:4) ! array to hold 5 coefficientsopen(unit=32.file.'D.dat'.access='seqout') ! output to b.datifl * 1 ! flag approximate b valuescall Fit(number~first.dpsi.b) ! get b valuer.close(unit=32)reoturiend
C-9
RHO-BW-CR-131 P
Subroutine Fit(numbersfirst1 dpsi.c)c Fit reads in 5 coefficients and uses them to approximatec either a or b. depending on whether it was called by Afitc or Bfit (in the first case. data goes to a.dat. in thec second. b.dat). The variable c is equivalent to eitherc a or b. whichever is appropriate.
dimension c(0:4) ! array of curve fit coefficientsdo 10 i-0.4 ! get them one at a timewrite(5 1)
1 format(t5.'Input a coefficient')read(5.2) c(i) ! read one in
2 format(f8.4)10 write(5.2) c(i) !write it to the terminal to confirmc calculate c and write it to filec unit 32 has been opened by our callerc as the correct output file (a.dat or b.dat)
psi first ! go through all psisdo 3 i = 1.numbertemp = 0do 4 n = 0.4
c the coefficients are for a polynomial fit inc psi squared4 temp = temp + ctn)*psi**(2*n)c the natural log of the data was used to determinec coefficients. so we must take the exponential ofc the result
result - exp(temp)write(32.5)result ! write to a.dat or b.dat
5y formatCf21.10)psi = psi + dpsi
3 continuereturnend
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RHO-BW-CR-131 P
Subroutine Ynice(number.amt)c Here we calculate y values with sleazy-but-nice formula.c (using only a and a user-supplied fudge factor). Actually.c we write out not Y itself but a close relative W (=b*ti).c The w values are put in (guess what?) w.dat.c define variables and open useful data files
real m,double precision y4bopen(unit=33file-'a.dat'.access='seqin')opentunit=34.file'b.dat'.access~'seqin')open(unit=35,file-'w.dat'.access=seqout')ifl 1 set flag: approximate y valuesDo 1 i 1.number ! 1 y for each psiread(33#2)a ! read in aread(34,2)b ! and b
2 format(f21.10)y = (t/a)**m ! m is fudge factorif((t/a).gt.1) y = Iw b*i ! get wwrite(35,2)w ! write w to w.dat
Subroutine Yexact(firstsdpsi.numberst~pi)c This routine gets exact values for y (and w)"c using a rather messy equation (equal flow only).c define variables and open files
double precision philtempl.temp2.w.gammaopen(unit=33,file'a.dat'aaccessin'seqin') ! a valuesopen(unit-34,file-'b.dat',access^'seqin') ! b valuesopen(unit=35.file'w.dat',access'seqout') ! put Ws hereifl = 0 ! flag exact y valuesDo 1 i = lnumber ! loop through all psisread(33.2)a ! get areadt34,2)b ! and b
2 formattf2l.10)phi pi*Cfirst+Ci-l)*dpsi) ! phi - pi*psiif( (t/a). gt. 1. O)goto 999 ! special case. w b
c invoke function to find gamma. used inc calculating g (w)
1 /(2*(sin(phi)-phi*cos(phii)))oldx s- xx = x + (oldf - f)/2 ! get nlotw xgoto 10 ! check again
20 Oappr - phi*xretuirnend
k I
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RHO-BW-CR-131 P
Function Cwft dpsi.number. pi. eps)c Approximate the integral for the concentration Cw
double precision tempoc.pi#w.aaac Oopen(unit=33Dfile='a.dat',access='seqin')open(unit=34.file='b.dat',access='seqin')open(unit=35,file~'w.dat',access-'seqin')do I i = Itnumberread(33.2) areadC34.2) bread(35.2) w
2 formatCf21.10)temp a exp(-((a-t)**2. ./ (4.*eps*w)))c - c + dpsi*temp/sqrt(4.*pi*eps*w)
SCIENCE APPLICATIONS, INC. TRACER TEST DATA IN BOREHOLES DC-7/8(reproduced from a report submitted by
K.I<_ Science Applications, Inc.)
K>
E-1
RHO-BW-CR-131 P
APPENDIX E
TRACER TEST IN BOREHOLES OC-7/8
The tracer test* was conducted as follows:
* With the shut-in tool open in both holes, pumping out of DC-7 wasstarted and continued for 2 days (before injecting the tracer mate-rial) at a discharge rate of ap2yt 1.3 gal/min. Approximately47.0 mCi of the water-soluble I tracer material were frozenusing dry ice. The frozen isotope was dropped (two small uncoveredplastic vials) into DC-8. Between 30% to 60% of the total dischargevolume out of DC-7 was injected into DC-8 until arrival of thetracer material at DC-7 is detected. The remaining water wasdiverted to a nearby pit.
* Following detection of the arrival of 131I at the pumping well,all water coming out of DC-7 was injected or recirculated intoDC-8. Triangle Service Company placed a 1 3/8-in.-outside-diametergamma detector tool at the wellhead of DC-7, and monitoring on a24-hr basis was conducted by Triangle personnel. This procedurehelped avoid contaminating tools, wireline, or other related equip-ment by radioactive material. Figure E-1 shows the surface equip-ment setup used in the tracer test. Pressure response in the twoholes was monitored continuously since the interval was straddledNovember 29, 1979 until the end of testing December 14, 1979.
TESTS, DC-7 AND 8/IF/9/TTO0
Data for the test are presented in Tables E-1 through E-4.
Figure E-2 shows part of the record obtained from the tracer experiment:Part A represents the background level; Parts B and C show the beginning andthe end, respectively, of the continuous rise of activity at the DC-7 welihead;Part D shows the steady flow of activity.
The activity (counts/inch) versus time in hours is shown in Figure E-3.Another graph is obtained from it by plotting the activity versus elapsedtime since injection as shown in Figure E-4. The tracer material was detectedat the OC-7 wellhead after 8 hr following its injection at DC-8. Theactivity increased sharply and peaked within <1 hr after it was firstdetected.
*The test yielded a value of 2.5 x 10-2 ft2/s for the product nb, whereb = 49.8 ft. The dispersivity obtained from the test is 1.1 ft.
E-2
C 'C
DETECTION EQUIPMENT NOT 70 SCALE
~METER
1. DETECTOR a
10Sf 1 9o5 | s <8.4 ft
- 249 f PIEZOMETRIC SURFACE - -- x -
01~~~~~~~~~~~~~~
5 in. 298.5 ft 1.6 in.
- ~~~~~~~~~3.03 In--.PUMP2.87 In. f
1 ~~~~~~~~46.2 ft
2.44 in. PACKERS
8.66 s | / 49.8 ft PERMEABLE ZONE
DC-7 DC-B
FIGURE E-1. Schematic of Tracer Test Setup.'.
RHO-BW-CR-131 P
TABLE E-1. Site Log for Test DC-7 and 8/IF/z9/TTOlTracer Test.
S - S 5-
STATUS ACTIVITY TIME DATE BY
PREPARE FOR TRACER TEST 12:01 12/08/79 KGK
DC-7 POMP ON 12:09 12/08/79 JB
START TRACER TEST DC-7&8/IF/9/TTO1 15:00 12/11/79 AAB
DC-8 DROP TWO FROZEN VIALS CONTAININGRADIOACTIVE MATERIAL (IODINE-131) 15:05 12/11/79 WS
DC-8 CIRCULATE: PART OF THE WATER PUMPED 15:16 12/11/79 WSOUT OF DC-7 IS INJECTED INTO DC-8
DC-7 TRACER MATERIAL SHOWING UP IN THE 22:57 12/11/79 AH* ~~PUMPED WATER225 12179 A
DC-7 CIRCULATE (INJECT) ALL WATER PUMPED 23:05 12/11/79 AHOUT OF DC-7 INTO DC-B
FIGURE E-4. Bredkthrough Curve of 131I.at Pumping Well (DC-7).
.
RHO-BW-CR-131 P
The time of travel of the tracer from DC-8 to DC-7 is actually the sumof three traveltimes:
1. Traveltime from the water level at DC-8 (about 140 ft below groundsurface) to the midpoint between the packers; i.e., a distance ofabout 3,280 ft
2. Traveltime in the formation itself, a distance of about 46.2 ft
3. Traveltime from the midpoint between the packers at OC-7 to theradiation detector at the surface, a distance of about 3,410 ft.
Thus, the tracer materialthrough 46.2 ft of formation.active material in IF/9 is cut
travels through 6,690 ft of pipe and onlyTherefore, the actual traveltime of radio-down considerably.
The various calculations leading to values for the effective porosityand dispersivity of the basalt interval tested are shown in Table E-5. Themethod used to arrive at these results is a modification of the theory.developed in Gelhar and Collins (1971) and is presented in the sectionentitled Traveltime and Dispersion Analysis.
The following summary tables for the discharge flow rate of pumping outof DC-7 and injection flow rates into OC-8 are obtained from Table E-4.
TABLE E-S. Average Discharge Flow RatesDuring Pumping Out of DC-7.
Date Time Interval Total time Average Flow Rates (gal/min)(hr) (min) Q div. Q inj. Q total
12/11/79 1505 - 1516 11 1.1 1.33 2.43
1516 - 1605 41 1.1 o.95 2.05
1605 - 1608 3 1.1 0.91 2.02
1608 - 1634 26 0 2.31 2.31
1635 - 2305 390 1.12 2.42 3.54
From the above tableOC-7 is 3.31 gal/min.
the weighted average discharge flow rate out of
E-28
RHO-BW-CR-131 P
Table E-6 shows a summary of the volume ofthe volume of water diverted to the nearby pit,out of DC-7 throughout the tracer test.
water injected into DC-8,and the total volume pumped
TABLE E-6. Summary of the Various WaterVolumes During the Tracer Test
Time interval Volumes of water (gal)
Date (hr) Injected Diverted Total pumpedinto DC-8 to pit out of DC-7
12/11/79 1505 - 2305 1,118
1516 - 2305 518 1,636
2300 - 2347 183 6 190
12/11/79 2305 to
12/12/79 14:45 6,164 _ 6,164
Therefore, the ratio (B in the section entitled Traveltime and DispersionAnalysis) of water volume injected to volume pumped is 0.68; i.e., theaverage injection flow rate into OC-8 prior to the arrival of the tracer is2.25 gail/min.
Traveltimes of the tracer material through the tubing in DC-7 and DC-8are obtained from the formula:
Ua (El)
where
u a average velocity of fluid moving in the tubing, ft/s
Q a injection (DC-8) or discharge (DC-7) flow rate, ft3/s
A - cross-sectional area of tubing, ft2.
For the traveltime of the isotope in DC-8 we use:
Q1 = 2.25 gal/min
A = tr2 = : (21;671)2 = 0.014 ft2.
E-29
RHO-BW-CR-131 P
Thus, the velocity, u, becomes:
2.25 (27.48-x 60 x 0.014 = 0.36 ft/s. (E2)
The distance traveled = 3,415 - 140 a 3,275 ft. Therefore, the tracermaterial takes 2.53 hr to travel from the water level to the top packer.This time becomes 2.55 hr when the distance traveled ends at the midpointbetween the packers.
For the traveltime of the tracer in DC-7 we use:
Qo = 3.31 gal/min
/2.441 2A - it = i 0.0325 ft2. (E3)
Thus the velocity, u, becomes:
U 3.31 ~~=0.227 ft/s. U7-.-48 x 6-0 x 0.0325 ' -2 t/.(4
The distance traveled - 3,414 - 257 1 3,157 ft. Therefore, the isotopetakes 3.86 hr to travel from the top packer to the pump. This time becomes3.91 hr when the distance traveled starts at the midpoint between thepackers. The tracer takes about 0.34 hr to travel from the pump through the1-in.-diameter tubing to the surface, a distance of about 257 ft. Thus, thetracer material spent 8 hr traveling through the tubing in DC-8 and DC-7,leaving about 1.2 hr for it to travel through 46.2 ft of formation. Thelatter time (1.2 hr) should be corrected for the differences in flow ratesfollowing the arrival of the tracer (see discussion following Equation E35).This is done below.
The traveltime, tp, of the peak in activity occurred at 23:30, wheretp = observed time of arrival of peak - time spent traveling through tubingin DC-8 and DC-7 - injection time.
To correct observed time of arrival of peak due to variations in flowrate:
23 + 6.8 x 0.5 = 24.03 hr (E5)
tp a 24.3 - 6.48 - 15.08 = 2.15 hr.
E-30
RHO-BW-CR-131 P
*The activity reached half its peak level at 23:15.
t C 0.25 x 6.8 = 0.51 hr. (E6)
Using Equation E22
nb = . 1
nb - 2.15 x 3,600 x 3.31(46.2)2 x 2ir x 448.8 x 0.2
nb 2.1 x 10-2 ft. (E7)
Using the thickness of the straddled interval b * 49.8 ft, we get aneffective porosity of 4.3 x 10-4. It is preferred to use the product nb,however, since the flow may be taking place through a much smaller sectionof the formation.
Using Equation E35, the dispersivity is:
= 0.3L t)05
= 0.3 x 46. x 0.78 ft. (E8)
Thus, the dispersivity of the straddled interval-is 0.78 ft.
DISCUSSION
The tested interval is about 220 ft above the top of the Umtanum flowof the Grande Ronde Basalt. The interval was effectively isolated by thepackers in both wells. The packers and electronic equipment performed ade-quately throughout testing.
No static pressure testing was conducted during this testing sequencein the interval. Results of static pressure testing performed earlier(between July 4 and 10, 1979) in the interval has been reported separately.Hydraulic head values obtained from the earlier static testing for formationconditions were 419.7 and 418.6 ft (mean sea level) for DC-7 and DC-8,respectively.
E-31
RHO-BW-CR-131 P
Transient testing consisted of one slug injection test, one slug with-drawal test, one air lift pump test, three pump tests, and one tracer test.Data from a DST conducted when the interval was straddled earlier were alsoanalyzed and gave a transmissivity of 6.3 x 10-5 ft2/s for the interval.The slug injection test of DC-7 resulted in a transmissivity of 6.6 x 10- ft2/s.No transmissivity or storage coefficients were calculated from the air liftpump test conducted at DC-8 with DC-7 as an observation well.
In preparation for the tracer test at this double well site, a submersi-ble pump was lowered at OC-7 to a depth of about 257 ft below ground surface.Pumping out of DC-7 continued for P48 hr. At the end of this period, develop-ing of both wells by intermittent pumping was initiated and, thus, the intervalpressure was not allowed to recover to static conditions. However, pressurerecovery data in DC-7 for the period December 5 through 7, 1979 were anal ze<.Transmissivities calculated from the first two pump tests were 1.3 x 10-: ftz/sfor DC-7 and 1.2 x 10-5 ft /s for DC-8. A storage coefficient value of1.1 x 10-5 was obtained from the observation well, DC-8, data. The thirdpump test gave identical transmissivity values for both wells, 7.8 x 10-5 ft2/s.The observation well, DC-8, data gave a storage coefficient value of 6.0 x 10-6.
A double well recirculating tracer tes was conducted in this interval.Approximately 47 mCi of the water-soluble I311 tracer material were frozenand dropped into DC-8. Pumping DC-7 at a rate of 1.3 gal/min had been initi-ated 2 days earlier. About 60% of the total volume of water pumped out ofDC-7 was recirculated into DC-8 until arrival of the tracer was detected atDC-7. The remaining water was djyerted to a nearby pit. A breakthroughcurve for the concentration of 1311 at OC-7 was constructed and used todetermine the effective porosity and dispersivity of the interval. The tracermaterial was detected at OC-7 after about 8 hr following its injection atDC-8. The activity increased sharply and peaked within <1 hr after it wasfirst detected. Note that the I11 tracer traveled through over 6,690 ft ofpipe and only through 46 ft of basalt (see Fig. E-1). The method used toarrive at these results is a modification of the theory developed in Gelharand Collins (1971). A value of 2.1 x 10-2 ft was obtained for the productnb, where n = effective porosity and b = interval thickness. The dispersivityof the straddled interval is 0.78 ft. These values agree with recent calcula-tions for the same test data.
It is of interest to compare our field-determined longitudinal disper-sivity value of 0.78 ft obtained for this formation with similar publishedresults. Unfortunately, field dispersivity values for fractured basalts arenot available in the literature. Grove and Beetem (1971) obtained disper-sivity of 125 ft from a tracer test using the recharge/discharge well pairmethod in a fractured carbonate aquifer near Carlsbad, New Mexico. Note,however, that the distance between the two wells was 125 ft at the surfaceand 190 ft at the depth of the formation; i.e., the dispersivity they obtainedis of the same order of magnitude as the spacing between the wells tested.Also, the method presented in their work differs from the method followed inour field test, in that the discharge flow rate from the pumping well intheir test was equal to the injection flow rate into the observation well.In our test only part of the water pumped out of DC-7 was injected into DC-8.Thus, comparing the two results may not be appropriate and is only meant togive an idea of the different dispersivity values available in general.
E-32
RHO-BW-CR-131 P
In a study using digital modeling to predict radlonuclide migration ingroundwater in basalt and sedimentary formations, Robertson (1974) foundthat a dispersivity of about 300 ft led to the calibration of his model.Thus, the dispersivity of this interflow zone is very small in reference todispersivities presented above. Recent work by Gelhar et al. (1979) showsthat the dispersivity is directly proportional to the distance, r, betweenwells. Figure 3 in their work indicates that the dispersivity will continueto increase with r until the spacing, r, is increased by a few orders ofmagnitude, and then it approaches an asymptotic value. For instance, thedistance between OC-7 and DC-8 would have to be more than 3,000 ft before anear-constant dispersivity for IF/9 can be obtained.
TRAVELTIME AND DISPERSION ANALYSIS
The following analysis is based on the theory developed in Gelhar andCollins (1971). The subsequent analysis applies to the test period from thetime of injection of the tracer material into OC-8 until it was first detectedat the OC-7 wellhead. During this time the discharge flow rate, Q0, out ofOC-7 was not equal to the injection flow rate, Qi, into DC-8. The relation-ship between the two can be expressed as (see Fig. E-5):
Q.; SzQ0 (E9)
Qo Qi
L
DC-7 0 0N DC-8
FIGURE E-5. Definition Sketch for Discharge/InjectionSet Up at DC-7 and DC-8.
E-33
RII0-BW-CR-131 P
where B = a constant. Assuming quasi-steady radial flow, the specificdischarge at some distance, x, along the shortest streamline between thewells is:
q (x) - q0 + qi (E10)
where
0 2irb r - L-x
and
qi 2nxb (Ell)
Figure E-6 is a sketch of the flow lines and the movement of the tracerfront in this case (i.e., qi J qo.). The seepage velocity, v, is definedas:
v (x) =9P(-xIn
(E12)
where n = effectivegives:
porosity. Substituting Equation Ell into Equation E12
To check the above equation, let us now evaluate It for the case whereB 0 and x = L; i.e., the radial flow case:
T (2 _) L - L2nb (E17).
This is the traveltime for the radial flow case. The case where B = 1at x = L; i.e., the discharge-recharge well pair when the flow rate, Qs isthe same, can be checked by evaluating Equation E14 using the followingexpression for velocity, v(x):
v(x) A (L- + I A ( L5HT (E18)
Thus
x 2 1 K L -___C _u -- u I du(E19)-v~x) AL ALu2 I
E-36
RHfO-BW-CR-131 P
For x = L, this reduces to:
LI'T W 6AirL2nb
C-Q0
(E20)
This is the traveltime for discharge/recharge well pair.time can be rewritten as follows:
The travel-
T= X- F(L; B)
where the function, F (E; B), is defined as:
(E21)
F (X; e) - I -,) x - 0 In (IL T, --S-) I L
B L,6 .
1
(1-B) 3
- 26 (1-B) + i2 ln
[((i-) X + 0 2 _
I + iji . x
Values of F (b; B) for r a 1 range from 1/6 to 1/2 for B values from I to 0.
E-37
RHO-BW-CR-131 P
The dispersion function, w, neglecting molecular diffusion,in Equation 22 of Gelhar and Collins (1971) as:
is defined
(E22)f t~
W
0
dx
IV(X)J
where x(t) is the location of the front at time, t, and is obtained fromEquation E21 by setting T - t. Substituting the velocity from Equation E13into Equation E22 leads to:
A2 [EL+(1-0)uI du (E23)
Upon performing the above integrations we obtain:
w A a 3 2 (1-0)z - 2BL(1-8) L
- 2L([ +(I-$) ]2
aL
-(0L) /2--
2 2 ~1X) - ( L L+(1-08jT - IlWEj
3BL(l-B)x + 302L2 In Il1}
+ 83L3(i.. 1TL
1(1-8)5
[([sLscl-86] - (L) /3 - 2(L rL+(1-B)x] - (81)2)
+ 6B2L (L-0)7
- 4(tL)3ln (I + El * L ) (BL)4(BL.(1-,l .L)]
The above expressiot. can be reduced to:
(E24)
w-zI
G (x; a) (E25)
E-38
RHO-BW-CR-131 P
where the funition G( ; B) is the right-hand side of Equation E24 afterdividing by L .
If B = 0 and the distance of the frontby F = L - x, then Equation E24 reduces to:
from the pumping well is given
2 - -2+ 3
which is the expression for thesolution, Equation 25 in Gelhar
- -1 (L 3_ 3)13- (E26)
radial flow analysis. The general pulseand Collins (1971), where n * T-t is:
C mP(0
I x~~(E27)
Notewhere is a constant, and a is the longitudinal dispersivity.0 u S0)that because of the large traveltime up to the detector, the pulse peak willhave moved through the aquifer before the tracer reaches the detector andthe flow is increased by complete recirculation. Equation E27 can bewritten:
L. Z.cm NIWUOt exp - w ] (E28)
where Cm - CAt - T and w0 - w(t)/t=T. We assume w0 = w(t) which means thatif the tracer peak is narrow then dispersion of the front as it travelsthrough an observation point is small. The peak concentration of the tracerat the pumping well occurs at time, tp, given by Equation E21 as:
P -r(L) I -. F(l;o) (E29)
where B = 0.68 for the test at DC-7 and 8/IF/z9 and A = Q0/2wnb.Substituting the value of A in the above expression we obtain:
nb *)L2
QO 12s 0.2
(E30)
where 0.2 is the value of the function F(1;B) for B = 0.68.
E-39
RHO-BW-CR-131 P
To calculate the dispersivity, we measure the time difference, At,defined as:
at = T(L)-t (E31)
when theEquation
concentration is half its value at the peak.E28 with w(t) = wo,
Therefore, from
- a exp I- r(-ti
or
1 exp - Ct)
Taking the natural logarithm of both sides gives:
(E32)
in 2 = (At)te val 4aw
Using the value of w from Equation E25. we obtain:
(E33)
(E34)
2
II 1n2 IL G ( I; :)
where A a
Thus,
- F(1;B) from Equation E21.tp
aI 1L 41n2
F2 2T-
(0.2)2 /A_41n2(.05) tp
fi r 0 3 -(&ta)IC V (E35)
E-40
RHO-BW-CR-131 P
Note that At is the actual time for the concentration to change from halfits maximum value up to the maximum.
It is of interest to note that the increase in the flow rate as thepulse gets to the detector causes the pulse to be swept by faster; i.e., insmaller time. We correct for this simply by expanding the time scale afterthe time of arrival (2300 on December 11, 1979) by the ratio Q3/Q2. where Q3is the flow rate during complete recirculation (after 2300 hr) and Qn = Qois the pumping rate from the well (OC-7) up to the time of arrival of thetracer.
REFERENCES
Gelhar, L. W. and M. A. Collins, 1971, "General Analysis of LongitudinalDispersion in Nonuniform Flow," Water Resources Research, Vol. 7,No. 6, pp. 1511-1521.
Gelhar, L. W., A. L. Gutjahr, and R. L. Naff, 1979, 'Stochastic Analysis ofMacrodispersion In a Stratified Aquifer," Water Reources Research,Vol. 15, No. 6, pp. 1387-1397.
Grove, 0. B. and W. A. Beetem, 1971, 'Porosity and Dispersion ConstantCalculations for a Fractured Carbonate Aquifer Using the Two WellTracer Method,' Water Resources Research, Vol. 7, No. 1, pp. 128-134.
Robertson, J. B., 1974, "Application of Digital Modeling to the Predictionof Radioisotope Migration in Groundwater", Proceedinas of IsotopeTechniques in Groundwater Hydrology, Vol II, International AtomicEnergy Agency, Vienna, Austria, pp. 451-477.
E-41
RHO-BW-CR-131 P
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I I I4.
DOUBLE-POROSITY TRACER-TEST ANALYSIS FOR INTERPRETATIONOF THE FRACTURE CHARACTERISTICS OF A DOLOMITE FORMATION
1INTERA Technologies, Inc.6850 Austin Center Boulevard, Suite 300
Austin, Texas 78731
2 Sandia National LaboratoriesAlbuquerque, New Mexico 87185
Abstract
Hydraulic and tracer testing studies have been performed as part ofthe regional hydrologic characterization of the Waste Isolation PilotPlant (WIPP) site in southeastern Ilew Mexico. The interpretation ofhydraulic tests and tracer tests from a number of locations has indicatedthe importance of the double-porosity concept in the interpretation ofboth the hydraulic and solute-transport characteristics of the fractureddolomite under study. While identification of fracture flow andestimation of formation transmissivity can be obtained from hydraulic-testinterpretation methods, tracer tests are necessary to provide realisticestimates of parameters such as fracture porosity and representativematrix unit size. Without estimates of these parameters, predictions ofcontaminant transport in fractured rock systems are very uncertain.
A convergent-flow tracer test conducted in the Culebra DolomiteMember of the Rustler Formation at the WIPP site was analyzed using adouble-porosity flow and transport model. The tracer test set-upconsisted of one pumping well and two tracer-addition wells arranged in anapproximate equilateral triangle with 100-ft (30-m) sides. The transportof conservative tracers fran each of the tracer-addition wells wassimulated using the double-porosity flow and transport model SWIFT II.The simulation model accounts for advective-dispersive transport in thefractures and diffusive transport in the matrix. Calibration of thetracer-breakthrough curves included conducting a sensitivity analysis onthe important parameters: diffusion coefficient, tortuosity, matrixporosity, fracture porosity, effective matrix-block size, pumping rate,
j.
I
initial tracer input distribution, and distance between pumping andtracer-addition wells. Calibration of the tracer-breakthrough curvesresulted in longitudinal dispersivities from 5 to 10% of the flow pathlength (well-separation distance), a fracture porosity of 0.002, andeffective matrix-block sizes of 0.8 ft (0.25 m) to 3.9 ft (1.2 m). Eventhough these estimates of fracture porosity and effective matrix-blocksize are specific to the location of the tracer test, they provide aninitial estimate on which to base predictions of the regional transport ofsolutes in the Culebra dolomite.
1.0 Introduction
Hydraulic and tracer-testing activities have been performed as partof the regional hydraulic characterization studies for the Waste IsolationPilot Plant (WIPP) site in southeastern New Mexico. The WIPP site islocated approximately 26 miles (42 km) east of Carlsbad, New Mexico(Figure 1). The site characterization studies are being coordinated bySandia National Laboratories on behalf of the Department of Energy andinvolve the evaluation of the suitability of bedded salt of the SaladoFormation for isolation of defense transuranic wastes.
The Culebra dolomite is the most transmissive rock unit above thewaste-emplacement horizon (Mercer, 1983) and for this reason it has beenthe main focus of site-characterization studies at the WIPP site. TheCulebra dolomite in portions of the WIPP-site area is considered to be afractured rock possessing both primary and secondary porosity (Rehfeldt,1984; Chaturvedi and Rehfeldt, 1984; Beauheim, 1987 and Kelley andPickens, 1986). The hydraulic and tracer-testing methods andinterpretation approaches have allowed quantification of fracture flow andtransport properties.
The H-3 hydropad is a well nest composed of three wells, and islocated in the south-central part of the WIPP site approximately 3900 feet(1190 m) south of the repository waste-handling shaft (Figure 1).Transport-parameter characterization of the Culebra at the H-3 hydropad isconsidered important because (1) the hydropad is located on a potentialflow path fran a repository breach under natural ground-water flowconditions, and (2) the hydropad is a potential site for theimplementation of a sorbing-tracer test (Pearson et al., 1987). Bothhydraulic and tracer tests have been interpreted using double-porositymodels at the H-3 hydropad (Beauheim, 1987; Kelley and Pickens, 1986).The double-porosity interpretation for a conservative-tracer test ispresented here with the interpretation reviewed for its consistency withthe physical system. The suitability of hydraulic and tracer tests todetermine appropriate double-porosity flow and transport parameters forthe Culebra dolomite at the H-3 hydropad is also discussed.
2.0 Site Characterization
2.1 Hydrogeology of the Culebra Dolomite Member
The sediments underlying the WIPP site range in age from Ordovicianto Recent. The sediments that are of most interest for characterizing theperformance of the WIPP repository are of Permian age and were laid down
146
[I
Figure 1 Site Location for the Waste Isolation Pilot Plant Showing theHydropad Observation-Well Network for Regional HydrogeologicCharacterization
149
in a deep-water embayment of the Permian depocenter known as the DelawareBasin. The Salado Formation is a thick-bedded salt section within whichthe waste repository will be located. Immediately above the SaladoFormation is the Rustler Formation, which is divided into five membersbased on lithology (Vine, 1963). Of these five, the Culebra DolomiteMember is considered to be the most transmissive unit (Mercer, 1983).
The Culebra dolomite at the H-3 hydropad is 23 ft (7 m) thick and islocally fractured. Interpretations of a 1 4-day pumping test in 1984 and a62-day pumping test in 1985 have ielded transmissivities of 3.0 and 1.8ft2/day (3.2 x 10- and 1.9 x 10 m 2/s), respectively (Beauheim, 1987).Beauheim (1987) also concluded that the Culebra responded hydraulically asa double-porosity medium at the H-3 hydropad and that the pumping wells inboth tests were intersected by fractures which substantially increased theproduction surface available to each of the wells. The hydraulic gradientunder undisturbed conditions (befqre shaft construction) at the H-3hydropad is estimated to be 4 x 10- to the south-southeast (Haug et al.,1987). Ground water from the Culebra at the H-3 hydropad has a densityequal to 64.8 lbm/ft 3 (1038 kg/m3) (INTERA, 1986). A modeling studyproviding a regional evaluation of ground water flow in the Culebra at theWIPP site is presented by Haug (1987).
2.2 Characterization of the Fracture Properties of the Culebra
Little quantitative information is available concerning the geometryof fractures in the Culebra, although several publications discuss theirpresence. From a review of some articles which do offer a degree ofquantitative information (Ferrall and Gibbons, 1980; Black et al., 1983;Holt and Powers, 1984; Rehfeldt, 1984; and Core Laboratories, Inc.,1986a), the following general conclusions were made: (1) both high-angleand horizontal fractures are present; (2) fracture apertures up to 0.3 cmwere observed within the original ventilation shaft at the WIPP site (nowthe waste-handling shaft); (3) fracture lengths from centimeters to amaximum of 2.1 m were observed in the original ventilation shaft; (4) whenfilled, fractures are most commonly filled with gypsum, yet in some casesthey are lined with oxides, pyrite, or bitumen; and (5) fracture facesexamined in core analysis possess surface textures and mineralizationindicative of fluid movement.
The range of fracture spacings to be utilized in modeling studiesshould be derived from information that is as site specific as possible.The best source of site-specific information is the description of thecore obtained from two of the three Culebra wells at the H-3 hydropad.Core recovery at the H-3 hydropad was very poor (Figure 2). Therefore,one must be discerning drawing conclusions from such a limited database. Approximately 10 percent of the Culebra interval was recovered inthe coring of the borehole designated H-3b2 and approximately 40 percentof the Culebra interval was recovered in the coring of the boreholedesignated H-3b3. At the H-3 hydropad, greater than 50 percent of thefractures are observed to be open in the recovered core. Both horizontaland vertical fractures are present. There are no recovered pieces of corelonger than 1 ft (0.3 m) in length and the core appears very porous.Because of potential core destruction during coring, estimation of anupper matrix-block size is not reasonable, however it is felt that
Figure 2 Results of Core Examination of Culebra Dolomite from BoreholesH-3b2 and H-3b3
the minimun matrix-block size can be estimated for the portions of theCulebra with recovered core. Through review of core and core photographs,0.5 ft (0.15 m) is considered a reasonable lower limit for matrix-blocksize.
3.0 H-3 Hydropad Tracer Test
3.1 Well Configurations and Downhole-Equipment Assemblies
A convergent-flow tracer test was conducted at the H-3 hydropadfrom May 16 to June 13, 1984 by Hydro Geo Chem, Inc., under contract toSandia National Laboratories. The H-3 hydropad consists of three wells,H-3bl, H-3b2, and H-3b3, arranged in an approximate equilateral trianglewith 100-ft (30.5-m) sides. A borehole deviation survey was performed atthe H-3 hydropad and the distances between boreholes at the Culebra depthare: H-3b1 to H-3b2 equal to 91.3 ft (27.8 m); H-3b2 to H-3b3 equal to87.9 ft (26.8 m); and H-3b3 to H-3b1 equal to 100.6 ft (30.7 m) (Saulnieret al., 1987).
Figure 3 shows the downhole configuration of the three H-3hydropad wells during the tracer-testing sequence. In the pumping well(H--3D3), a submersible pump was installed belao a Baski air-inflatable
151
sliding-end packer, with the discharge line extending through the packerand then to ground surface. A packer and feed-through plug were also usedin the H-3b2 tracer-addition well during the tracer test. The packersystem also served to minimize the system volume during tracer addition byisolating the wellbore fluid in the annular space above the test intervalfrao the downhole system volume (see Figure 3). Fluid pressure wasmeasured in each borehole with Druck PDCR-10 pressure transducers.
Tracers were injected into wells H-3bl and H-3b2 fran thesurface through 1/2-inch (1.3-an) polyethylene tubing. For H-3bl, thepolyethylene tube was lowered inside the 2-3/8-inch tubing string to adepth of 525 ft (160 m) below top of casing, where it encountered anapparent obstruction preventing positioning of the tube to a lowerdepth. In well H-3b2, the 1/2-inch polyethylene injection tube was fedthrough the packer feed-through plug to allow injection to the zone belowthe packer.
3.2 Tracers
The tracers used in the H-3 hydropad tracer tests, pentafluoro-benzoate (PFB) and meta-trifluoromethylbenzoate (m-TFMB), are anhydrousacids derived fran benzoic acid. PFB has been tested extensively, both inthe field and in the laboratory, and has not shown evidence of sorption or
H- 3bi H-3b2 H-3b3
Top of C01,6933S9? It 6.1.. (1033.1s nil
Total depth 902 ft. (274 9 m)
Figure 3 Schematic Representation of the Downhole Configuration at theH-3 Hydropad During the Tracer Test
11
-
�i.V
.egradation (Stetzenbach, personal communication). m-TFMB has also been-ested extensively and shows no signs of sorption. Experiments carriedzuL at the University of Arizona have shown it to be resistant to-agradation for at least six months (Stetzenbach, personal.- mmunication). Hydro Geo Chem (in preparation) used five fluorinated'anzoic acid tracers in tracer tests performed at another location at the;IPP site and concluded that only m-TFMB and PFB showed no signs of..egradation. These two tracers have the additional strong points of being.1) exotic to the Culebra ground water at the H-3 hydropad, and(2) detectable at very low concentration levels through proven analyticallachniques (Stetzenbach et al., 1982).
3.3 Tracer-Test History
The H-3 hydropad tracer test was a oonvergent-flow tracer test inWhich well H-3b3 was pumped at a nominally constant rate of 3 gpm'0.19 Vs), while 2.2 lb (1 kg) of tracers m-TFMB and PFB were injected!vith 100 gal (379 1) and 60 gal (227 1) of formation fluid, respectively,over a 1.6-hour period into wells H-3bl and H-3b2, respectively. Thebreakthrough curves obtained for the m-TFMB and PFB tracers fran watersamples from the pumping well H-3b3 are shown in Figure 4, and the arrivaltimes, first-measured and peak concentrations, and percent tracer recoveryat the pumping well are summarized in Table 1.
I
I_12
WEC.0LU
E
0I-6I-
I-zLuUz0U
4000
3000
2000
1000
0.00 .
& .&m-TFMB
0-a PFB
- A
AA
A~~~
A~ ~~~~~~~~A...... . .....A~~~
0 10.0 20.0 30.0TIME SINCE TRACER INJECTION (DAYS)
40.0
Figure 4 Tracer Concentrations at the Pumping Well forExpressed as Micrograms per Liter
PFB and m-TFMB
153
First detection of m-TFMB (reported concentration of 56 Ug/l) at thepumping well was obtained from water samples taken 22.1 hours (0.92 days)after tracer injection at well H-3bl. The maximum observed concentration(3379 ug/l) was recorded 62.08 hours (2.59 days) after tracer injection.By integrating for the mass below the m-TFMB breakthrough curve over theduration of the test, it is estimated that approximately 53 percent of theinjected tracer mass was recovered at the pumping well. First detectionof PFB (reported as trace concentration) at the pumping well was obtainedfrom water samples taken 90.25 hours (3.76 days) after tracer injection atwell H-3b2. The maximum observed concentration (444 wig/l) was recorded553 hours (23.04 days) after injection (Figure 4). By integrating for themass below the PFB breakthrough curve over the duration of the test,recovery of approximately 15 percent of the injected tracer mass isestimated. The PFB breakthrough curve exhibits a peak concentration abouta factor of eight lower than the m-TFMB breakthrough curve and a time toreach the peak concentration delayed by a factor of nine when compared tothe m-TFMB breakthrough curve.
3.4 Interpretation Approach
The objectives of the interpretation of the H-3 tracer test were todevelop a consistent conceptualization of the governing physical solute-transport processes operating in the Culebra at the H-3 hydropad and todevelop quantitative estimates of the respective transport parameters.
From a review of the information base for the H-3 hydropad, it wasconcluded that a double-porosity interpretation approach was the mostappropriate. This information base included: (1) identification of openfractures in core samples; (2) very rapid transport rate between the
Table 1 Summary of Tracer Arrival Times and Mass Recoveries at thePumping Well H-3b3
TracerParameter m-TFMB PFB
Flow Path H-3bl to H-3b3 H-3b2 to H-3b3
First reported concentration 56 20(i/1)
Time of first detection (days) 0.92 3.76
Time of arrival of peak 2.59 23.04concentration (days)
Peak concentration (ug/l) 3379 444
(M/M) 2.9 x 10-6 3.5 x 10-7
Tracer mass recovered 53 15during the test (%)
-:a&r-addition wells and the pumping well; and (3) the identification of:2;ure "low and a double-porosity pressure response from the analysis of:^: lumping test at the H-3 hydropad (Beauheim, 1987).
Because of the relatively high matrix porosity of the Culebra, solute- -ar-sport between the fractures and the matrix by diffusion is expected to-e ^ significant process. Therefore, a discrete fracture model with:-asport.in the fractures only is not considered an appropriate concep-.: a.ization for analyzing the H-3 tracer test. The double-porosity.:,roach is considered appropriate and is described below.
The concept of a double-porosity medium was first proposed by?arenblatt et al. (1960) to model flow in fractured rock. Inherent to the..rincept of a double-porosity medium is the idea that the medium consistsof two separate, interacting and overlapping continua. Using theStreltsova-Adams (1978) classification for dual-porosity reservoirs, the-ulebra at the H-3 hydropad is a class-one dual-porosity reservoir which'.s termed a fractured medium. In a fractured medium, the primary medium'the matrix) has the greater porosity and effectively represents the bulk,:f 'one "storage" capacity of the unit, and the secondary medium (theIractures) has "transport" properties generally as a result of secondaryprocesses (i.e., post-depositional). Also inherent in double-porosityt9heory is the concept that any representative finite volume of thecollective media contains both primary and secondary media.
In a double-porosity medium, various assumptions are necessary toallow the system to be represented mathematically. One very importantassumption is that the system can be characterized as fractures and matrixunits with a relatively simple interaction between them. The Culebra isassuned to have three orthogonal fracture sets. This fracture/matrixsystem is modeled as spheres. Spheres are advantageous because the cubicgrid geometry is awkward for modeling internal diffusion. This problem issolved by approximating the cube matrix units by spheres having the sanesurface-to-volume ratio as a cubic block (Neretnieks, 1980; Rasmuson andNeretnieks, 1981; Rasmuson et al., 1982). Neretnieks (1972) found thatthis approximation yields the equivalent uptake as cubes for short timesand only varies slightly for larger times. While these mathematicalidealizations cannot be expected to represent natural geologic systemsexactly, they do allow the solution for double-porosity transport at thefield scale to be a tenable problem. Conceptually, one should considerthese ideal representations as approximations of the natural system, whereone is attempting to attain quantitative consistency between the fracturefluid volume and the surface area available for diffusion.
Further assumptions were made regarding the conceptual basis for thisanalysis. The flow field in the study area was assumed to be radialaround the pumped well and at steady-state conditions during the tracertest. Since the hydraulic conductivity of the matrix is low and the flowregime is approximately at steady state, advective transfer from thefractures to the matrix and advective transport in the matrix were assumedto be negligible. Therefore, the transport of the tracer from thefractures to the matrix and within the matrix was assumed to occur bydiffusion only. The double-porosity medium was assumed to be homogeneous,and isotropic.
I 511-
0
The SWIFT II model was selected for simulation of the tracer testsperformed at the H-3 hydropad. SWIFT II is a fully-transient, three-dimensional, finite-difference code capable of solving the coupled equa-tions for flow and transport in a double-porosity mediun. A comprehensivedescription of the theory and implementation of the SWIFT II model ispresented in Reeves et al. (1986).
The transport equations were solved in a radial coordinate system.A Cartesian coordinate system would be impractical because of the verylarge number of grid blocks and time steps that would be required toprevent numerical problems. Additional reasons for choosing the radialapproach to simulate the H-3 tracer test are: (1) it offers advantages inmeeting the numerical criteria of the model, and (2) the field data baseon heterogeneity is not sufficient to warrant utilizing the Cartesianapproach, which is much more difficult to implement.
A schematic representation of the global discretization in both planview and cross section is shown in Figure 5 for the pumping well and atypical tracer-addition well. The pumping well resides at the center ofthe radial system and is given a constant discharge rate consistent withthat measured during the tracer test. Both upper and lower boundaries of
(te Cross Section Pumping Well Tracer-Addition Well
- Outer
Edgeof
Modeled
Region
VbI Ppln View
Outer Edgeof
ModeledRegion
Figure 5 Schematic Representation of the Modeled Region, (A) CrossSection, (B) Plan View
i56
I11
system are considered to be no-flow boundaries. At the outer edge of:.3 ,adial system, a Dirichlet pressure boundary condition is
-- :scvibed. Using the radial simulation approach, the initial tracer.ribution surrounding the tracer-addition well is approximated as being
.3ributed in a concentric ring surrounding the pumping well. The actual'A tial tracer-input zone at the end of the short-duration tracer-_,action phase and the approximated zone for modeling purposes are showninematically in Figure 5. Further details on the calculation procedures
estimating the mass of tracer introduced into each of the global gridcks at the input zone are outlined in Kelley and Pickens (1986).
The radial discretization implemented for the global system assumes.Iat the geologic system is both homogeneous and isotropic. As a result,law paths for tracers m-TFMB and PFB were analyzed separately. Results'rom anisotropy determinations from pumping tests performed at the H-4,
i-5, and H-6 hydropads at the WIPP site (Gonzalez, 1983) have yielded_.nisotropy ratios of 2.7:1, 2.4:1, and 2.1:1. This degree of anisotropyfs weak in magnitude and cannot clearly be differentiated fron the effects
aquifer heterogeneity. Although it is felt that hydropad-scale._terogeneities are present at the H-3 hydropad, the quantification of the
-patial variability of the medium properties is not possible frcm theexisting information base.
1.5 Model Input Parameters
As discussed earlier, the Culebra dolomite at the H-3 hydropad isconceptualized as a double-porosity system for solute-transport modelingpurposes. Parameters characterizing the Culebra, tracers, tracer-testoperating conditions, fractures, and matrix were utilized in fitting then-TFMB and PFB breakthrough curves determined from water samples taken.rom the pumping well. The following is a discussion of the estimation ofvalues for each of these parameters.
Tracer Free-Water Diffusion Coefficient
Free-water diffusion coefficients for m-TFMB and PFB have beenreported by Walter (1982). He calculated the free-water diffusion coeffi-cients using the Nernst expression and data from laboratory experimentsconducted to determine the limiting ionic conductances of the tracerspecies. The calculated diffusion coefficients for m-TFMB and PFB were6.9 x 10-4 ft /d (7.4 x 10-° cm2/s) and 6.7 x 10-4 ft 2 /d(7.2 x 10-6 an2/s), respectively (Walter, 1982).
Tortuosity
The solute diffusion coefficient in the porous matrix is defined asfollows for use in the SWIFT II code:
*D = Dm tDo (1)
where D is equal to the solute molecular diffusivity in the porousmatrix; is equal to the matrix porosity; X is equal to the tortuosity;and D i~? equal to the free-water diffusion coefficient. In studyingsoluteP transport by diffusion, tortuosity is a parameter whose
1I57
magnitude (0< 'r <1 ) is a measure of the tortuous nature of the poresthrough which the solute is diffusing. As the resistance to diffusionaltransport for a conservative species increases, the magnitude oftortuosity decreases. Bear (1972) has presented a review of tortuosityvalues in the range of 0.3 to 0.7. Bear states that tortuosity iscorrectly defined as:
X . (L/Le)2 (2)
where L is equal to the straight-line distance; and Le is equal to themean length of the diffusional path in a porous matrix.
Although not stated, Bear's review appears to have been for studiesutilizing unconsolidated media. Reported tortuosity values forconsolidated materials like dolomite are rare. Tortuosity values of 0.02to 0.17 were calculated fran the diffusion coefficients of Cl in chalksamples by Barker and Foster (1981). Fran diffusion experiments oncrystalline rock samples, Katsube et al. (1986) calculated tortuosityvalues of 0.02 to 0.19. It is expected that the tortuosity will varyspatially within the Culebra. For simulation purposes in interpreting thetracer test at the H-3 hydropad, tortuosity values of 0.15 and 0.45 werechosen.
Longitudinal Dispersivity
In fitting tracer-breakthrough curves for transport in a single-porosity medium, longitudinal dispersivity is often the key parameterutilized in the calibration (i.e., fitting breakthrough-curve shape andpeak concentration). In a double-porosity system, the transport ofsolutes between the fractures and matrix by diffusion can have a verylarge effect on the breakthrough curve, thus causing the interpretation ofthe best-fit longitudinal dispersivity to be difficult. A review of theliterature on the magnitude of longitudinal dispersivity for varioustracer-test scales and contamination-plume sizes (e.g., Lallemand-Barresand Peaudecerf, 1978; Pickens and Grisak, 1981) suggests that longitudinaldispersivity can be expressed as a function of the mean travel distance ofthe tracers or contaminants. In many situations, the longitudinaldispersivity is from 5 to 10 percent of the travel-path length. Since thewell spacings at the H-3 hydropad were approximately 100 ft (30 m),longitudinal dispersivities of 4.9 ft (1.5 m) to 9.8 ft (3.0 m) werechosen for simulation of the breakthrough curves.
Fracture Porosity
Fran examination of the tracer-breakthrough curves, it was concludedthat the rapid arrival of the m-TFMB tracer at the pumping well could havebeen dominated by transport in fractures along the H-3b1 to H-3b3 flowpath. A first estimate for the fracture porosity was calculated from therelation
- QE / i r2h (3)
where is equal to the fracture porosity; Q is equal to the dischargerate atf the pumping well; t is equal to the time to reach the peak
Iconcentration; r is equal to the distance between the tracer-addition and-pu-niing wells; and h is equal to the aquifer thickness. This equation isbased on the assumption that transport is occurring in the fractures onlywith no tracer losses to the matrix. Therefore, it will yield anoverestimate for the fracture porosity.
Using equation 3, the calculated fracture porosity based on thei-W:MB breakthrough curve for the H-3b1 to H-3b3 flow path was approxi-
miately 2.0 x 10-. The PFB breakthrough curve for the H-3b2 to H-3b3 flowpath exhibited a much later first detection of tracer and did not have a4ell-defined peak concentration. Therefore, only the fracture porositydetermined frcm the m-TFMB breakthrough curve was utilized. Because ofthe large difference between the breakthrough curves for m-TFMB and PFB,it is recognized that the estimated fracture porosity is uncertain and itsrepresentativeness to both flow paths may be questionable.
Matrix Porosity
Porosity determinations conducted on six core samples fran boreholesH-3D2 and H-3b3 ranged from 0.11 to 0.24 (Core Laboratories, 1986b) withan average value of approximately 0.2. A matrix porosity of 0.2 waschosen for simulating the tracer-breakthrough curves.
Matrix-Block Size
As discussed in Section 3.4, the conceptualization of the double-porosity system involves mathematically representing the natural system asa homogeneous, idealized configuration of fractures and matrix units. Formodeling the tracer-breakthrough curves at the 11-3 hydropad, the matrixunits were assumed to be defined by three orthogonal fracture sets. Bothhorizontal and vertical (or near vertical) fracture sets have beenobserved in core samples, shaft excavations, and outcrop areas. Eventhough the natural system is heterogeneous, one must attempt to develop areasonable approximation of the correct fracture fluid volume and thesurface area available for diffusion from the fractures to the matrix.
A spherical representation of the matrix units was chosen forsimulation purposes. Because the time scale of the tracer tests is notvery long, with the depth of penetration of the tracer into the matrixunits not large, the spheres are mathematically equivalent to the cuberepresentation through a consistent correlation of fracture fluid volumeand surface area available for diffusion. This assumption in thesimulation approach is discussed further by Kelley and Pickens (1986).
The characteristic matrix-unit size (i.e., fracture spacing) isexpected to vary considerably over the WIPP-site area and also verticallyat any location as a result of the high degree of heterogeneity observedin the Culebra. Matrix-block sizes from 0.5 ft (0.15 m) to 3.3 ft (1.0 m)are considered a reasonable range based on examination of core samples.These matrix-block sizes were utilized as initial estimates for simulatingthe tracer-breakthrough curves using the SWIFT II model.
Pumping Rate
The discharge rate at the pumping well was relatively constant at3 gpm (0.19 1/s) throughout the tracer test. This pumping rate was usedto simulate the tracer-breakthrough curves.
Culebra Thickness
The definition of the bottom and top of the Culebra has been reviewedusing available geophysical logs (Beauheim, personal communication). AtH-3b1 the Culebra thickness is 24 ft (7.3 m), at H-3b2 it is 23 ft (7.0m),and at H-3b3 it is 23 ft (7.0 m). A Culebra thickness of 23 ft(7.0 m), corresponding to the thickness estimate at the pumping well, waschosen for simulating the tracer-breakthrough curves.
Distance Between Tracer-Addition and Pumping Wells
Distances between the boreholes at the Culebra depth were calculatedbased on the surveys of the borehole locations at ground surface andborehole-deviation surveys (Saulnier et al., 1987). The distances betweenH-3b1 and H-3b3 for the m-TFMB flow path and H-3b2 and H-3b3 for the PFBflow path are 100.6 ft (30.7 m) and 87.9 ft (26.8 m), respectively.
Initial Tracer Input-Zone Dimensions
The tracer-test history was presented in Section 3.3. The tracer-injection procedure consisted of mixing the tracer in an initial volume ofwater, injecting the tracer-labeled volume, and injecting a second volumeof water to displace the tracer-labeled water into the formation. Sincethe injection was of short duration, it was assumed that the tracer movedout under plug-flow conditions through the fractures only and resulted inan initial tracer input zone that was cylindrical in shape and encompass-ing a region dependent on the volume injected and the fracture porosity.The two fluid volumes that are injected determine the initial tracer-zonedimensions in the aquifer. Natural gradients were assumed to have anegligible effect on the initial tracer-mass distrgibution around theinjection well. For a fracture porosity of 1.9 x 10' (determined duringthe model calibration of the m-TFMB breakthrough curve) and the respectivefluid volumes injected, the initial tracer-input ring of the m-TFMB tracersurrounding well H-3bl had inner and outer radii of 3.3 ft (1.0 m) and5.6 ft (1.7 m), respectively, and the PFB tracer surrounding well H-3b2had inner and outer radii of 4.9 ft (1.5 m) and 5.6 ft (1.7 m),respectively.
The input parameters discussed above are represented by a constantvalue for each simulation during calibration of the breakthrough curves.However, input-parameter values for different simulations are adjustedsystematically, within ranges judged as reasonable, in an attempt to matchthe observed tracer-breakthrough curves.
3.6 Analysis of Tracer-Breakthrough Curves
Fran initial inspection of the two breakthrough curves (Figure 4),one can identify major differences in tracer breakthrough. The m-TFMB
160
;urve peaks sharply early in the test, whereas the PFB curve is veryr'oad, is of much lower concentration, and requires a significant portion:. the test period to reach the maximum observed concentration. With the:.--rrent double-porosity conceptualization, it was possible to achieveeasonable breakthrough-curve matches with system parameters consistent
.iith the current physical and conceptual understanding of the Culebra..le best-fit parameters are summarized in Table 2.
The tortuosity chosen during calibration has a direct effect on the*stimate of matrix-block size because tortuosity is part of the product:nhich SWIFT II defines as molecular diffusivity (see Equation 1).2asmuson and Neretnieks (1981) define the characteristic time fordiffusion as:
to ' Lm2 /D (4)
where t is equal to the characteristic time f9F diffusion; L is equal toone-half of the matrix-block length; and D is equal tom the solutemolecular diffusivity of the matrix. This characteristic time fordiffusion in part controls the transient behavior of the breakthroughcurve. Therefore, if one varies tortuosity (i.e., D*), then one mustCaompensate with the value of m to achieve the same breakthrough curve.Good fits between observed and simulated breakthrough curves were obtainedfor tortuosities of 0.15 and 0.45. Fran a literature review oftortuosities for consolidated materials, 0.15 is considered to be moreappropriate. Only the results for a tortuosity of 0.15 are presentedhere.
Figure 6 shows the comparison of the observed and simulatedbreakthrough curves (concentration expressed as mass per unit mass) forthe m-TFMB tracer on the H-3b1 to H-3b3 flow path for a tortuosity of0.15. A longitudinal dispersivity of 9.8 ft (3.0 m) and a fracture
Table 2 Summary of Best-Fit Input Parameters for m-TFMB and PFBBreakthrough Curves at the H-3 Hydropad
TracerParameter m-TFMB PFB
Solute free-waterdiffusion coefficient (ft2/d) 6.9 x 10-4 6.7 x 10-4
(m2/s) 7.4 x 10-10 7.2 x 10-10Tortuosity 0.15 0.15
Matrix-block length (ft) 3.9 0.8(m) 1.2 0.25
Longitudinal dispersivity (ft) 9.8 4.9(m) 3.0 1.5
Fracture porosity 1.9 x 10-3 1.9 x 10-3
Matrix porosity 0.2 0.2
I.,I
4. OE-06
3. OE-06 - O uArveo mft-ms Concentration3.9 ft. 12 ml Simulated m-TFM8 Concentration
IN
2 .mlOEi Oz0
2.OE-06
I-zwU
0
i. OE-0eA AI
0.0.0 10.0 20.0 30.0 40.0
TIME SINCE TRACER INJECTION (DAYS)
Figure 6 Observed and Simulated Breakthrough Curves for Tracer m-TFMB
porosity of 1.9 x 10-3 provided the best-fit simulated breakthroughcurve. An effective matrix-block size of 3.9 ft (1.2 m) is consideredrepresentative for fitting the m-TFMB breakthrough curve. Simulatedbreakthrough curves for matrix-block sizes of 4.3 ft (1.3 m) and 3.6 ft(1.1 m) are also shown for comparison in Figure 6.
The observed and simulated breakthrough curves for PFB for theH-3b2 to H-3b3 flow path are shown in Figure 7 for a tortuosity of 0.15.As discussed earlier, the breakthrough curve for PFB did not indicatefracture-controlled transport. Using the same fracture porosity as forfitting the m-TFMB breakthrough curve, a dispersivity of 4.9 ft (1.5 m)and an assuned tortuosity of 0.15 resulted in an effective matrix-blocksize of 0.8 ft (0.25 m).
a A simulation was conducted using SWIFT II with a single-porosityconceptualization for the H-3b1 to H-3b3 flow path (m-TFMB flow path) toevaluate whether or not a single-porosity model would adequately simulatethe tracer-breakthrough curve. The parameter values were chosen similarto those for the double-porosity analysis of the m-TFM breakthroughcurve. A tortuosity of 0.15 and porosity of 1.9 x lo 3 were chosen. Thisporosity was chosen because it has the primary control on the arrival timeof the peak concentration. The observed m-TFMB breakthrough curve and thesimulated breakthrough curves for the single-porosity and double-porosity
i~~~~~~~~~~~~sI,
3. OE-07
[I
B. OE-07 Observed PFB Concentration a
Simulated PFS Concentration-
0- 4. OE-07 Fracture Spacings 0.9ft. 0 °
2.0E<7~~~~~~402 in)
0.0~~~~~~~. 10. 20.25 in 00.
0
2. OE-07-
0.00.0 i±0.0 20.0 30.0 40.0
TIME SINCE TRACER INJECTION (DAYS)
figure 7 Observed and Simulated Breakthrough Curves for Tracer PFB
conceptualizations are shown in Figure 8. It is evident fram comparisonof the observed and simulated breakthrough curves that it would bedifficult to obtain a single-porosity calibration of the observed tracer-breakthrough curve.
Dispersivity is defined and applied consistent with a Fickianconceptualization in the SWIFT II model. The longitudinal dispersivitiesobtained for the two flow paths are within a factor of two. Fran a sensi-tivity analysis (Kelley and Pickens, 1986), it was found that increasingdispersivity alone caused an earlier tracer arrival and higher peakconcentration. The difference in dispersivities for the two flow pathscan be viewed as a measure of either differences in heterogeneity betweenthe two flow paths traveled by the tracers or a result of difficulties inproviding a unique fit of observed and simulated breakthrough curves forprocesses described by such a large number of parameters.
The characteristic matrix-block sizes estimated francalibration of the breakthrough curves varied between the flow paths. Theestimated block size for the H-3bl to H-3b3 flow path is approximately4.8 tUmes larger than the block size estimated for the H-3b2 to H-3b3 flowpath. Although the calculated matrix-block sizes are consistent withobservations from core samples, shaft excavations, and outcrop areas,there are various factors which might in part explain the different
16.3
4.OE-05
3. 2OE-05
N
-2.OE-05
Single -Porosty .
0~~~~~~~~0 Oud Io -or Os; i_ I
0.-
0.0 10.0 20.0 30.0 40.0TIME SINCE TRACER INJECTION {DAYS)
Figure 8 Ccmparison of Observed and Simulated Breakthrough Curves (Single-.and Double-Porosity Conceptualization) for Tracer m-TFMB
matrix-block sizes calculated for the respective flow paths. Same of the <possible factors are: (1) differences in tracer behavior (diffusion'coefficients, sorptive characteristics, biodegradability); (2) differencesin tracer-input conditions (i.e., input functions); (3) anisotropic andnon-haxogeneous transport characteristics of the medium; and (4) regionalgradients being -of significant importance relative to local gradientsinduced by the convergent-flow field.
Differences in tracer behavior have been shown to be minimal inlaboratory free-water diffusion tests (Walter, 1982). Both tracers m-TFMBand PFB have been used for convergent-flow tracer tests in the Culebradolomite at the H-6 hydropad. If one did suspect that these two tracershad different behavior under field conditions, one would expect theirdifferent behaviors to be similar at various tracer-test locations. Incontrast, at the H-6 hydropad, m-TFMB had a very broad and lcw-concentration breakthrough curve while PFB had a very sharp and relativelyhigh-concentration breakthrough curve. This implies that the breakthroughcurve differences stem from flow-path differences rather than tracer-behavior differences. As mentioned in Section 3.2, both tracers shouldhave been resistant to biodegradation over the duration of the test(Stetzenbach, personal communication).
I'~~~~
164
.'s mentioned earlier, another potential cause for the differences in: akthrough curves is a difference in input functions at the two tracer-:.->ion wells. If all of the tracer mass did not enter the aquifer
-. ,iallv, tracer could have continued to have been introduced to theAifer f'cm the well during the test. This condition would result in a:: eeendent input function that could have an important effect on
.;erpreting the breakthrough curves. After each of the tracer-ections, a sufficient volune of water was added in order to displace, welibore volumes completely within the first 1 to 2 hours after tracer
*..iection. The input functions were simulated to match this input-ocedure. Due to a lack of information concerning wellbore concentration.Ksus time in the tracer-addition wells, and due to the non-uniqueness of
.. e calibrated parameters, it is not possible to assess whether sane of-.. e tracer mass may have remained in the tracer-addition wells.
The model treated the Culebra as an isotropic homogeneous medium. It- known that hydropad-scale heterogeneites exist at the H-3 hydropad.Irougn core reconnaissance. Anisotropy has been determined to be weak
zm hydraulic testing at other hydropads and cannot be differentiated-am heterogeneities (Gonzalez, 1983). Through the analysis of two long-;rn pumping tests conducted at the H-3 hydropad, Beauheim (1987) found
*-evidence for a preferential hydraulic connection between wells H-3bl and-. b3. If this is true, and the H-3b1 to H-3b3 (m-TFMB) flow path is more
:irect than the H-3b2 to H-3b3 flowpath (PFB), this could explain the-roader and lower concentration breakthrough curve exhibited by PFB. Thiseffectively means that the PFB tracer would travel a longer flow path, andsince this extension of the flow path is not accounted for in the model,the matrix-block size must be decreased to compensate for the apparentincrease in diffusivity or the real increase in surface area available for"iff usion.
Regional hydraulic gradients range from 1 x 10-3 to 4 x 10-3 in theCulebra (Mercer, 1983). Hydraulic gradients at the H-3 hydropad duringthe convergent-flow tracer test are estimated to be an order of magnitudegreater than the regional gradient and, therefore, would have dominatedlocally.
It is recognized that uncertainty exists in the assumed orcalibrated values for tortuosity, fracture porosity, matrix porosity, andmatrix-block size used to describe solute transport at the H-3 hydropad.Reducing this uncertainty would require additional laboratory and fieldtesting (e.g., additional drilling and coring, additional matrix-porositydeterminations on core, diffusion experiments, and additional field tracertesting). The results obtained from the conservative-tracer test indicatethat fracture flow and matrix diffusion dominate solute transport in theCulebra at the H-3 hydropad. Further, the parameters derived to fit them-TFMB and PFB breakthrough curves are thought to be consistent withcurrent conceptualizations of the Culebra at the H-3 hydropad.
4.0 Conclusions
The following conclusions can be drawn from interpretation of aconservative-tracer test conducted at the H-3 hydropad in the CulebraDolomite Member at the Waste Isolation Pilot Plant site:
1. The rate of transport of the tracer between the tracer-additiorwells and the pumping well indicated that transport was dominatecby the presence of fractures and by diffusive transport between thEfractures and the matrix.
2. The tracer-breakthrough curves could be simulated by approximatingthe Culebra dolomite as a fractured medium with three othogonalfracture sets. While this is an idealized representation of anatural system, it is consistent with other physical observationsand provides the ability to handle the solution of double-porositytransport at the field scale as a tenable problem.
3. The interpreted matrix-block sizes for the two tracer flow paths atthe hydropad were 3.9 ft (1.2 m) and 0.8 ft (0.25 m). Because ofthe large number of fitting parameters, uncertainty exists in therepresentativeness of these matrix-block sizes. However, theyshould be considered to be at least qualitative and to provide anindication of the fracture-fluid volume and surface area availablefor diffusion in order to fit the observed tracer-breakthroughcurves.
The estimates of fracture porosity and effective matrix-block sizeare specific to the location of the tracer test. They do provide,however, initial estimates on which to base predictions of tracertransport for a proposed sorbing tracer test and for regional scaletransport of solutes in the Culebra.
Acknowledgements
This work was supported by the U.S. Department of Energy under contractDE-AC04-76DP00789. This study has been improved by technical review fromG. E. Grisak and G. J. Saulnier (INTERA Technologies, Inc.) andA.R. Lappin, and D. Tanasko (Sandia National Laboratories). The authorswould also like to thank C. H. Lowenberg and J. E. Cramer for their carein preparing this manuscript.
Biographical Sketches
Both J.F. Pickens and V.A. Kelley are hydrogeologists with INTERATechnologies, Inc., at Austin, Texas. Research interests include theapplication of hydraulic and tracer testing techniques for characterizingthe transport properties of porous and fractured geologic environments.
Mark Reeves is a senior staff consultant at INTERA Technologies, Inc., andholds three degrees in physics. He is perhaps best known for hisdevelopment and application work with two groups of flow and transportcodes: the saturated-unsaturated models now known as FEMWATER and FEMWASTEand the nuclear waste isolation models SWIFT and SWIFT II. Currently, heis working on a group of new models called SYSNET which statisticallyanalyze human-intrusion scenarios for a nuclear waste salt-siterepository.
166
_ eauheim is a hydrogeologist with the Earth Sciences Division of::ia National Laboratories. His background has involved the development
application of field and interpretive methodologies for characterizing:-tured geologic formations. He currently directs the hydrologic
:.:escigations at the Waste Isolation Pilot Plant site.
..ferences
: 'enblatt, G.I., Y.P. Zheltov, and I.N. Kochina, 1960. Basic concepts inthe theory of seepage of hanogeneous liquids in fissured rocks(strata). Journal of Applied Mathematics and Mechanics (USSR), Vol.24, No. 5, p. 1286-1303.
3BrKer, J.A. and S.S.D. Foster, 1981. A diffusion exchange model forsolute movement in fissured porous rock. QJ. Eng. Geol., V.14,p. 17-26.
sarn, J., 1972. Dynamics of Fluids in Porous Media. American ElsevierPublishing Company, New York, 764 p.
Leauheim, R.L., 1987. Analysis of Pumping Tests of the Culebra DolomiteConducted at the H-3 Hydropad at the Waste Isolation Pilot Plant(WIPP) Site. Sandia National Laboratories, SAND 86-2311.
Black, S.R., R.S. Newton, and D.K. Shukla, editors, 1983. Results of SiteValidation Experiments - Volume 2 of 2. Supporting Documents 5-14.U.S. Department of Energy, TME 3177.
Chaturvedi, L. and K. Rehfeldt, 1984. Groundwater occurrence and thedissolution of salt at the WIPP radioactive waste repository site.American Geophysical Union, EOS, July 3, 1984, p. 457-459.
Core Laboratories, Inc., 1986a. A Canplete Petrographic Study of VariousSamples From The Rustler Formation. Performed for INTERATechnologies, Inc., under contract for Sandia National Laboratories,Unpublished.
Core Laboratories, Inc., 1986b. Special Core Analysis Study for INTERATechnologies, WIPP Site, File Number: SCAL 203-850073. CoreLaboratories Inc., Aurora, Colorado.
Ferrall, C.C. and J.F. Gibbons, 1980. Core Study of Rustler FormationOver the WIPP Site, Sandia National Laboratories, Contractor Report,SAND79-7110, 80 p.
Gonzalez, D.D., 1983. Groundwater Flow in the Rustler Formation, WasteIsolation Pilot Plant (WIPP), Southeast New Mexico (SENM): InterimReport. Sandia National Laboratories, SAND82-1012, 39 p.
Haug A., V.A. Kelley, A.M. LaVenue, and J.F. Pickens, 1987. Modeling ofGround-Water Flow in the Culebra Dolomite at the Waste IsolationPilot Plant (WIPP) Site: Interim Report, Sandia NationalLaboratories, SAND86-7167.
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Haug, A., 1987. A Modeling Study of the Culebra Dolcmite, paper presentat the NWWA Conference "Solving Ground Water Problems with Model.held February 10-12, 1987, Denver, Colorado, 26 p.
Holt, R.M., and D.W. Powers, 1984. Geotechnical Activities in the WastHandling Shaft, Waste Isolation Pilot Plant (WIPP) Project in Soutieastern New Mexico. U.S. Department of Energy, WTSD-TME-038.
Hydro Geo Chem, in preparation. Convergent Flaw Tracer Tests at the H-Hydropad, Waste Isolation Pilot Plant (WIPP), Southeastern New Mexic(SENM). Unpublished contractor report prepared for Sandia NationaLaboratories, 88 p.
INTERA Technologies, Inc., 1986. WIPP Hydrology Program, Waste IsolatiotPilot Plant, Southeastern New Mexico, Hydrologic Data Report #3.Sandia National Laboratories, Contractor Report SAND 86-7109.
Katsube, T.J., T.W. Melnyk, and J.P. Hune, 1986. Pore Structure FrarnDiffusion in Granitic Rocks. Atomic Energy of Canada Ltd., TechnicalReport TR-381 28 p.
Kelley, V.A., and J.F. Pickens, 1986. Interpretation of the Convergent-Flow Tracer Tests Conducted in the Culebra Dolamite at the H-3 andH-4 Hydropads at the Waste Isolation Pilot Plant (WIPP) Site, SandiaNational Laboratories, Contractor Report SAND86-7161.
Lallemand-Barres, A. and P. Peaudecerf, 1978. Recherche des RelationsEntre la Valeur de la Dispersivite Macroscropique d'un MilieuAquifere, Ses Autres Characteristiques et les Conditions de Mesure.Bull.Bur. Geol. Minieres (Fr.) Sect. 3, 4:277.
Mercer, J.W., 1983. Geohydrology of the Proposed Waste Isolation PilotPlant Site, Los Medanos Area, Southeastern New Mexico. U.S. Geo-logical Survey Water-Resources Investigation Report 83-4016, 1'13 p.
Neretnieks, I., 1972. Analysis of sane washing experiments of cookedchips, Sven. Papperstidn., 75, p. 819.
Neretnieks, I., 1980. Diffusion in the rock matrix: An important factorin radionuclide retardation?, J. Geophys. Res., 85, p. 4379.
Pickens, J.F. and G.E. Grisak, 1981. Scale-dependent dispersion in astratified granular aquifer. Water Resources Research, V.17, No. 4,p. 1191-1211.
Rasmuson, A. and I. Neretnieks, 1981. Migration of radionuclides infissured rock: The influence of micropore diffusion and longitudinaldispersion, J. Geophys. Res., 86, p. 3749-3758.
Rasmuson, A, T.N. Narasimhan, and I. Neretnieks, 1982. Chemical transportin a fissured rock: Verification of a nunerical model. WaterResources Research V.18, No. 5, p. 1479-1492.
168
.--eves, M., N.D. Johns, and R.M. Cranwell, 1986. Theory andImplementation for SWIFT II, Sandia Waste-Isolation Flow andTransport Model for Fractured Media, Release 4.84. Sandia NationalLaboratories, NUREG/CR-3328 and SAND83-1159.
.- hfeldt, K., 1984. Sensitivity Analysis of Solute Transport in Fracturesand Determination of Anisotropy Within the Culebra Dolomite. NewMexico Environmental Evaluation Group, EEG-27.
aulnier, G.J., Jr., G.A. Freeze, and W.A. Stensrud, 1987. WIPP HydrologyProgram, Waste Isolation Pilot Plant, Southeastern New Mexico,Hydrologic Data Report #4. Sandia National Laboratories, ContractorReport SAND86-7166.
Stetzenbach, K.J., S.L. Jensen, and G.W. Thompson, 1982. Trace Enrichmentof Fluorinated Organic Acids Used as Ground-Water Tracers by LiquidChromatography. Environmental Science and Technology, V.16,p. 250-254.
treltsova-Adams, T.D., 1978. Well Hydraulics in Heterogeneous AquiferFormations, Advances in Hydrosciences, V.T. Chow ed., V.II, New York,New York, p. 357-423.
Vine, J.D., 1963. Surface Geology of the Nash Draw Quadrangel, EddyCounty, New Mexico. U.S. Geological Survey Bulletine 1141-B, 46 p.
Walter, G.R., 1982. Theoretical and Experimental Determination of MatrixDuffusion and Related Solute Transport Properties of Fractured Tuffsfrom the Nevada Test Site. Los Alamos National Laboratories,LA-9471-MS, 132 p.
169
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iIIIIIiI
I
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i
i
II
ii
SOLVING GROUND WATERPROBLEMS WITH MODELS
An intensive three-day conference and exposition devotedeXclusiheelv to ground water modeling
FEBRUARY 10-12, 1987FAIRMONT HOTELDENVER, COLORADO
Sponsored byThe Association of Ground Water Scientists and Engineers
(A Division of the National Water %Well Association)International Ground Water Modeling Center, Holcomb Research Institute
.*Alas coltsboratsr at the LAS ALMOO stia21 Laboaiciy. LAs ALmm. -leg
I
lie soratton oi cesium ana strontium nas been moceled wtth aheterogeneit-aesea isotherm ecuation ior various cutf mstertalsincludirng those within a secuence oi geologic strntigrapnic units.Mhe theory of the isotherm ioresees the relative retarcation and the
,hrmscal dispersion" of the studied radionuclides during transport.The concepts of heterogeneity of sites and variability in theMtmiuM nrnrr oi sites available ior sorption are incorporated Intothe model.
7nROrrzw
olcaiic tuff material is tne geologic mecia ior tre snallow i--= burtia ot
:or-ievet raaioactive waste at t:eAs os £am0s iationll LxboratorT in nortn-
western New Amico. klso. volcanic tuff mvterial of Yucca Aotuntain near
south-central Nevada Is the proposed geologic medium for a repository oi
high-level radioactive waste (1]. kie aspect of the cheracierlution oi
either Gisposal arsa is the sorptive belavior of rodiamclides in the tuff
uterials. -his chracterization Is iortant in predicting waste migration.
Isotherm hays been derived and used to t re ent sorptire belavior in a
vartety of disciplines. e.g.., soil chemistry. geachestry. vrd ewirowmental
md chemicat engineering L2. -. -i. S. 31. -he Iao Ir Isotherm Is basec on
-o smolest of several tVeortes are available to aescribe tne
oeiationsrzdp between the amont ot a solute aasoroed on a surace ane the
lAncentration of the solute In the liquid piase (73. -his isotherm asumes
:hat the energy of adsorption Is the ss for all active sites on the
adsoroent surface. rowever. in now adsorption cases. :hs assiumtion fails
becauis either pure mineral or uiltt-sinerai suriaces Interaet with sonutes
-11th different energies Iheterogensity). 1ese cifferences among sit.,
-nergies reausre tne identification oi those energy distributions nat
zAracterize the heterogeneity of a particular nasoroent-solute interaction.
I
U
Sips LS] Introaucea and discussea an isotherm tnat -u~ests tnamt n
:aussian-iikc statistical function could represent the aistrtbutlon nt
itte-soiute interactions. 7his isotherm, which is basea on the nssumotton oi
.ocaltized adsorption without interaction among sites. was presented as an
cxp stcn of the cornentional Freundlich isotherm. Soosito (9]. following the
results catained by Sips (8] tuoa using a LaWzuir isotherm to define
site-solute interactions. derived a similar Gaussian-like statistical function
that is regarded as a log-noruni distribution oi a variable that defines the
deative affinity oti a sOaute tor a solid ohase.
'he isotherm can be exuressea as
max& I + K:' C
where
S a quantity of solute adsorbed per unit mass of solid phase i.
C1 a cencentration of solute in solution in equilibrium with solid phase
1.
3xt - meaxgmtn available exchange camncitv at solid phase i. an
_.' j u nerameters that define tne overall sciute-solid onase
Interaction.
r this isotherm applies. then a more ctorenensive representation oi the
.Aterogene ty of the adsorption is gained from the meaning of 3 3nm 0. ;iese
: o peraseters can be found by regrissional analysis ot a given set oa
Jorotion data on the following expressIon:
p_ lo C, tog ( c 2)
Ihe crmeter i has been cescribed by Socsato 19] as a measure at how
-narply peaked the statistical function tar ti;2 alstrabution ot energies at
-ouilibrium is naout an average value: the parameter ras also been aescr:bed
I
v Cricma~ore ana nlo$ciccnoswsxi -:1J as the snreno ot tiic iaLtcL
* .jr t- distribution ni aoisorationl-aesorvtion rate constant~.; aacr
L~as been i~i~citiv rieiatoo by ,;posito gJto an averaze :-.str,.-jt-cn
-oafficient." ;r an nverame nosorotion energy or vitfinit~v: Crickmorre ano
IoJciechoswski (C01. ,.n trte otner r-mid. def ine it as tz-.e ratio o: the r'action
.he cojectives ot th~is oaper are ii) to introduce onen give cnemical
ceaninig to the parameters o1 the general equation. %`) to verifv the Validity
of the isotherm azd to interpret isotherm parameters tar sorption otf strontium
vid cesumti on Banoelier ruff with resoect to their i--xct on tn-eir transport.
-ird (3) to extezd tie no~piicntion of the isotherm to str..tlirranic units
man~zterials with aifferent cavacities to axisorc radionuclides.
'ATERIALS AMD ITTROD
7 xaerimentai
1amie~ier 'ruff Is cescribed ~eoaogically as a voicanic asnl flow comooseo
-astkY am sillcic ziass with a grain size cistributlon ciose ra un'at oi :
~iity sana and a cation excnannge canpacity i(() oi 3.3 mmit fni*/t'.
'inerniogicaily tne tuif !3 primariiy comosea of qu~artz. .rldymite. aido
-ikall feldscars. '*tailed procecures and tocrinicues for trne 1eatcn aosorotion
xnertments are cnescrioea bv rnizer F~t at. ..e cescritintcl.d
-
proceaures of the laboratory coiumn transport studies are given by Fuentes eE
al. [12].
The tuffs ot Yucca Mountain have been described in mineralogic ana
petrologic studies (13]. The batch sorption procedures followec for the
Yucca Mountain tuff media are described by D[intels et al. f14].
The batch sorption studies of the tuffs from Yucca i(ountain were not
designed to evaluate Isotherms. but to evaluate the effect on adsorption oi
such parameters as contact time. concentration of sarbity elements. _article
;zie. :sa-erature. atmosonere. _aw lithoiogy. '%erefoy2. :nie ranee in
Concentrations varied from one exDersment to anotner. aly cata ior - contact
time of 14 days. ambient temperature. and air atmosPhere were used In the
modeling effort. Sorption data Include those for samples of all ranges o:
particle sizes. Hmwever. particle sizes varied from on tuff sanle to
another. 7relitairny modeling (15) Indicates that available particle sizes
had little effect on Isotherm poaraeters.
2.2 Modeling
The data were modeled using the heterogtneit-tnsea isotherm (Xodifled
.'reundlich). Zata sets were £ine£riy regressen ana the results were acefinea
by coefficient of deteratnations (Rj. conmide I level of the regression
anlysits. coofficlent of variation (CV). and the values for the parameters as
determined from the slope and from the Intercept. Mhe intercept Is the
depenoent variable when the independent variable. concentration in solution.
In unity. The error terms associated with the parameter values are also
4ncluded. hes.e statistics and parameters provide criteria for Mooaness oi
isotherm fit and for estimates oi the extent of heteroceneity of the
a sorption.
I
I1
-' 2.SULTS ArD DISCUSSION
i. 2andelter ruff Faterial
The Modified Freunalich isotherm olts fcr the sorp.on of strontium ana
cesium to Bondelier i'zff are Presented in Fig. ;. The statistical Parameters
:or the regression analysis and the isotherm parameters are given in Table l.
The R- Is 0.993 for th2 cesium regression and 0.9S9 for strontium regression.
The coefficient oi variation is 7.3 for cesium and 2.7X for strontium. 7hese
statistical data indicate a very good correlation between tne fraction of
;orbetd sites occupied bv tne tracers i'S/(S Qil anm the concentration or
:racers iC) in solution. 7.e vaiues ior U were O.S42 for Ltrontium ano O.C62
.or cesium. 2sea on tne theory described. :.ose values suggest that tne
spread (dispersion) In the breaxthrougn (or transport) of strontium cue to
heterogenetty of sorption sites should be less than that of cesium. -his
spread should be in addition to the spread due to flow. The value ior
strontium is about one-half of that for cesium (0.172 and 0.308.
respectively). .;yatn. :ased on the above theory. those relative ivalues
indicate that the oreakthrough or transport of cesium snould be retarded bv
:oproximateiy a factor ci 2 coemarea with trat ox strontium.
.oe aoove results ana the theory are suoportec. at least cuaiitativeiv ov
zhe resuits ot laboratory coiumn breakthrougn data snown in Fig. '. Tlese
iata are acanted from ituentes (12]. -he relative breaithrouen pattern aopears
.o follow that precicted by the isotherm parameters for strontium ana cesium.
The breakthrough of Iodide represents transoort of a nonsorbing tracer and the
dispersion of iodide should be cue to flow conditions. 7he appearance oi the
tnttiai ana peak breaxthrougns oi strontium ana cesium indicates that czstum
:s retarded by ncout a mactor at 2 comoarea with strontium as orecictec from
he isotherm parameter. . so. .s sDrean In tr.e breaxthrouen anpears ti
I
,e much greater for cesium than tor strontium. ..awever. shoulid be rotea
.hst a strict interpretation must include an evaluation oi dispersion cue tn
inter flow. -he spreac due to alow dispersion is not tne same zor -ne two
tracers because ot differences in arrival or breaxthrougn times. £Ce aoove
results and conclusions appear to give validity to tne Modified Freundlich
isotherm as a means of predicting tte relative transoort of raoionucildes in
tuff natertals. 'lrk on the coupling of the isotherm eauatton to a transport
-odel is in progress.
3.2 Yucca Mountain Tuff Mtaterials
he results just aiscussea above are rialtec' in t.-at tne tutif oterial ;.is
zoilecteo from a specific area ai the &aricelier straticrapnic nitl. :o minimize
.nhomoeenities in characteristics. a.e. g tximm sorption capacity (CEC)V sn
order to extend the isotherm model to larme environmental areas, a variable
mluxrm sorption capecity needs to be Incorporatec into the model. 'or
exaple. within a stratigrapnic unit of Yucca Mountain available informltion
indicates that the (C= of the various eaterials ray vary by a factor of 60.
This incorporation was accoailished by Including a CEC (S ' for eacn set ofMaX-
orption cata. Thus the decendent vartable is comoosec of two variables. -he
zmount -o-red. S. ana the maximum sorption cavacitv. ! n tre form
3/(S *i). This proceaure was usec to evaluate tne appuicability °t the
Isotherm to model sorption for five stratigraphic units oi Yucca Mountatn
ocated below the canoidate repository. she results are presented in Fables 2
mand 3. ¾ble 2 gives the number of data sets used in the analysis. Lhe ranze
'n uaxiim sorption capacities (CEC). :he coetficient oi variation (CV). :no
the confidence levei of the regression analysis. ble 3 presents tne
-arameter estimates for trie i-ocrerm caseo on tne regression analysis. '!cure
' provides a `-orse case anoa est ase examole ot !orotion accoroinc to fro
I
Aodifled Freundlich isotherm. ..;e --orse' case is tne ony' yituation tor
tntch the Isotherm does not represent tne cata weil. - snotld be emnnasizca
as mentioned previously that the sorotion cata were not ceneratca for tne
eurpose of isotherm analysis. ;.nich may be a reason for imoroper nrcacalnZ
of the experimental values in the worse cases.
The statistics for the regression analysis indicaL - ef icient of
determination of 0.66 or better tor the regresston anai;:3 with one
exceptton: ;r was oniy 0.55 for the regresston of cesiu ata in the C-ilico
Mllls unit. The coefficient of variation of the data about tne regression
ine raozec from anout I -O L. .- e hieher C'/s were oniv a :actor "_;reater
than tCloSe deterainec with tne -anaelier ruff 7hus tnese cata suegest tfat
incorporation of S-M !nto the cependent variable does not greatly affect
variation of the data about the regression line.
The zuraaeter estimates are plotted as a function of the stratigraphic
unit In Fig. 4. 3ome general trends can be observea within the limits of
errors associated with the estimates. -he average K values for cesium and
for strontiua follow similar natterns. The log I'D varies from auproxIuateiy
-2.0 to *2.0 with the Calico Hills unit having tne nighest % _na the rrow
'ass having the lowest K. 1he g values range iroam oproxlmateiv 0. CS to
i.5. As mentioned earlier. ;ne J parameter is theoretically exuectea to be
less than 1; the higher value as 1 .S only occurs for strontium in the Calico
4ills unit. :n this set of observations the data are not uniformly
distributed: one observation extends the doaiin of coservations from
approxivately I order of ragnitude to about 3.5 orders of nagnitude. bi
influence diagnostic test based on D-Cook-like statistics f16] Incicates that
:he observation unculy influences tre recression ana the stone 4I). 1 tbe
Observation were to oe celetea. .ne slope would decreasr. .nus raintailnin tne
3 parameter within theoretical predictions. 7e cnemical cisoersion .'.r
aoth strontium and cesium appears to tollow a similar -attern oi chance as
rhat for the iC. but the pattern is less obvious. Significant differences
between the O's for cesium and those for strontiu occur in the Tram unit -na
perhaps in the Bullfrog unit. The similarities in the pattern of cnange ot
the IC 6alues and those os the ii vaiues suggest that when the iCD (average) is
large. .3 approaches unity. This relation liies that as the average K.
increases. the spread of individ'al Kv vatues about the average is less.
The iarce error associatea with parameter estimates oi the cesium tracer
.n tne Calico Hills unit is irnicative of the retativeiv iew caota available
aid oi their poor distribution over the narrow range of observatiOns (less
than I order of magnitude). -he influence diagnostic test basea on
D-Cook-like statistics (15] indicates that several sorption 6oservations
unduly influence the regression and parameter estimates. those sae
observations account for most of the relatively large residual resulting from
;he regression analysis. ven though the coefficient of determination is
relatively poor (0.55). a confidence level of greater than 96Z for the
-egression suggests a aefinite ilnearIty associateG with the regression tnus
xuuportinx the anpllcability oi the isothenm
The patterns of change of K- and 3-values between stratigraphic units
oecome a simplifled means to characterize the heterogeneity of the adsorution
of strontium and cesium in Yucca Mountain materials. The expected spatial
variability for the adsorptton process can be scopea from the meaning oi :
and P. rnaeiv, retardation Intensity (1%) and variability of retardation 40).
This result is an improvement over cre use oi conventionat isotherms that co
-ot recognize the potential heterogeneity ot aosorption. L.g.. ..angmuir. its
;eterogeneity couid be uncerstooa as a "chemical dispersion ?;ecause it mry
j
cause a difference in raoionuclide migration due to differences in
retarcation. These indings suggest a retter approacn to hancie aosorption in
transport modeling of :adionuclides in porous media. :n recomnizin: that the
data sets used In this study were not collected with this modeling ajpproach in
mind. there is a need to improve the data base to further test the response oi
the model.
ACNnOLE/DCKENT
We are indebtea to tee iievada Nuclear Waste Starage investigations
?roject for tne use of data on tne crption ot racionuclides to Yucca Mountain
Tuff materials. The data were coilected by the aluclear and Radiocnemistry
Croup. 1C-ll. at the present time under the supervision of Dr. i. W. Thomas.
'Fe are grateful to E. II. Essington for his help in the develpopent of the
batch and column exceriments with Bandelier Tuff. to C. Larghorst for computer
przgramniag support, and to Dr. R. J. Becknus for helpful discussions on
statistical analyses.
-EF=RNCS
: 1SH. D. L. VAArKN D. T. ;YERS. 3. ; t Lncr D. E. urmnrof the Hineralory-Oetroioqy of Tufts of Yucca Mountain andSeccndary-Fhases of Therni Stability in Tuffs. Los Alamos NatitonaiLaboratory Report UA-9321-MS (1982).
2. (XI.DBD~Q. S.. S.3lTO. C.: A chemical Model of Phosphate Adsorption bySoils: I. Referenco Oxide Matertals. Stoil Sci. Soc. Am. J. iL, 72(194).
3. TRAVIS. C. C.. ETNIER. E.L.: A survey of Sorption RelationshiDs forReactive Solutes in Soil. j. Environ. Quality .lO. 8 (19SI).
-4. OLFE. T. A.. Z!XIREL. T.. 3AUUANN. E. R.: Adsorption of OrnziicPollutants on Montmorillonite Treatec with Amines. aour. VPCF LS. 3( 1966) .
;IEM. W. j.. PIRAZARI. H.: 'Adsorption oi Toxic anu Carcinoqenicrompounds from Water. .aur. .kWTA BA. ..03 (1982).
6. fwFniN. D. .: Principles of Adsorption Pxd Adsorption Processes. cahnWiley and Sons. Inc.. :Nw York 1984. p. S6.
7. LAICWIR. I.: he Adsorption of Cases on Plane Surfaces oi Class. 3caand Platinum. J. A. he. Soc. 40. 1361 (1918).
S. nrs. 2.: On the Structure of a Catalyst Sarface. J. Chem. Phys. t64! ) (1948).
9. SPOFr. C.: Derivation of the Freundlich Equetion for ion ExcizangeReactions In Soils. Soil Scl. Soc. Am. J. i. 652 (1990).
10. CUXlD P. J.. T= ExmuKI. B. W.: Kinetics of Adsorption onEnergetically Heterogeneous &Srfaces. J. Chem. Soc. Faraday Trans. 1M.1216 (1977).
1l. .U.M. W. L.. WIMlTS. H. R.. INgCIU'i. E. H.. -dERSM. F. R.:Equilibriua Sorption of Cioblt. Cesium. and Strontium on aond lier ruff:Analysis of Alternative OathemeticaI Kodeling. 7aste *'(ametnt SS. _,:67 (19651}.
;2. iUriE. .J. R.. PaZER. J. L.. 3DINGalu . E. H.: Effects of Sorption andTemperature on Salute Transport in Uhsawurated Steady Flow. 'taste
lIngmelt as. a, 3t (196).
13. Meioi. D.. YANIDXA. D.. CAPMUSCI0. F.. AMr. 3.. inEDKE. C.:Dstailed Petrographic Descriptions end 9icroprobe Data for Drill HolesUSF-( and UE25b-IH. Yucca Nountain. N1evada. Los Alaos fatlenalLaboratory Report LA-932t-RS (1982).
It. DMIFLS. W. R.. ICSEERC. K.. * B Z . R. S.. CCMAZ. A. E.. .RSK, J.F.. DOM3FS. C. J.: Skumry Report on the Geochemistry of Yucca Mountainand Environs. Los Alamos Katiomai Laboratory Report LA-932S-MS (1982).
15. FATMES. H. R.. PCLZER. 1. L.. COMER. j.. LUAI 3ES. B.. 3r1MUoI. E. H.:Preolmazry report on Sorption Modeling. Los Alamos National LaboratoryReport LA-109=2-15 (19671.
:6. SAS InaMU1E MIC.: SAS Users Guide: Studies Version 6 Edition. LASInstitute Inc.. Cary. Nforth CarollA IS965. p. 676.
10
table 1. 3eatiuttcal am Modifiled Freumclich isothermparameters estimated from the regrezstoni armjysis of thesorption of cesius and strontium on Bandelier tuf' 08
Sr Cs
0.9994
cv (%)
Cmf idenceLevel (0.OIz)
2.7
0.9999
0.9925
7.3
0.9999
3
:533.9 i 0.010
.).172 i 0.00r
).662 i 0.014
'). 308 *k 0. 077
1)Oand P have smits correspondlag to S(p#.)/g and C In pmoo(pt)/d. In Eq. 2.
and Sma inl umol
II
I
;T1Ab 2. Experimental p.Araaeters for sorption of Ptiooasilimsi elli csUIum on tiaff mccll;:from stratgraphle units of the Yucca Mountain area and tatilstical parametersestfuated from regrresslo. a..alybi for the Modified Freundlich sorption Isotherms
Isporite Calemu.l
Prowbass Iziltrg Tro
_
I coSt Cs firc co sr' co ST co
la l b i I N 3 .t Idal
i,.nga oaIIWALaS12 Cuw&csmarmttmio(Pae1694I
a- .... GI S
3.CE02-2 I.3F2-0
b.*-333OC-O1ID-9to to
4.02-0? 4.02-09
W E-11 IaL X-CAe- to
6.02-0? 4.01-0
b LA.; Ui A C -Cto. t
2.CE-63 6.0E-09 3.41-0) 1.39-00
.1s AO '.eO La.j * 35*? 49 go 1300 I: ... I143 3A i. '69
G.ifig 0 Lo;, U 1)1 0.150 0.k-2 0.947 I L. Ii . 0.W13 O.W3
t
V I-) 1~~0 4.6 3.9 2.2 9.2 2.? 6.4 3.9 If to
C gCf`14AazCs (O.O3X) a.i*.i. O.Sttw. .O~A t. 0.06*1 O.9WAI 0.Oj.oo 0 9..-. 0.90gBo OAM O 0.w l
a*.9, ud jP have uani as cc a.:sporaidi to S and S., In pua~l(p+)/a, and C In jamol(p.)/aaJ.in Eq. 2.
I
.able 3. :odifled Freunalich isotherm parameters estimated fromregression analysis for the sorption Oi strontium and cesImeon tuff maeci from stratigrapnic units ci the Yucca Mountain nrea-
Tran 0.87 i 0.04 0.74 i 0.04 0.57 i 0.15 ).26 * 0.16
4KO and 8 have unlts corresponding to S and Sam in &oi(p+)/g and C Iny=ol(p+)/zL In Eq. 2.
I
I13
)
I
a
Cc.
Ila,
ESfo
'-d
01
I
I
I
iII
1*
t ;L1 -e!. -ib -1i5 -iC0 -.15G 10
log [C(urnoles(pi-Vni~1.O :z
es2
a
C2.
-I--I q
ea
C.
.2a
I.;
IiI
:2
a
, &
, a-i0.0-9LOlog (C(UMo~esdp+Y/MD.1
C.0 I.a
,7tq. . todlf led Freuzndlich isotherm plots of strontium (upper P and cesium(lower) on Landelier MuLf..ata are for scration atter -IS h a,equilibration am~ at a temeratureofc 23 i 3uC.
0.
0.01-
0:5 0
Fig. 2. 3reaktbrougan curves of reactive (strontium and cesium) andncreactive (iodide) tracers in a ilaboratory colutn of BandelierTuft at 25"C and under unsaturated flow conditlons.