Permeability and Hydraulic Diffusivity of Waste Isolation Pilot Plant ...

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Permeability and Hydraulic Diffusivity of Waste Isolation Pilot Plant ...

SANDIA REPORTSAND92–1911 l UC–721Unlimited ReleasePrinted June 1993--RECORD COPYC1 lMICROFICHE.Permeability and Hydraulic Diffusivity ofWaste Isolation Pilot Plant Repository SaltInferred from Small-Scale Brine InflowExperiments ,D. F. McTiguePrepared bySandia National LaboratoriesAlbuquerque, New Mexico 87185 and Livermore, California 84550for the Unlted States Department of Energyunder Contract DE-AC04-94AL85000II 1I*8608829*I IISANDIA NATIONALLABORATORIESTECHNICAL LIBRARYSF2900Q(8-81)


Issued by Sandia National Laboratories, operated for the United StatesDepartment of Energy by Sandia Corporation.NOTICE This report was prepared as an account of work sponsored by anagency of the United States Government. Neither the United States Governmentnor any agency thereof, nor any of their employees, nor any of theircontractors, subcontractors, or their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privatelyowned righta. Reference herein to any specific commercial product, process, orservice by trade name, trademark, manufacturer, or otherwise, does notnecessarily constitute or imply ita endorsement, recommendation, or favoringby the United States Government, any agency thereof or any of theircontractors or subcontractors. The views and opinions expressed herein donot necessarily state or reflect those of the United States Government, anyagency thereof or any of their contractors.Printed in the United States of America. This report has been reproduceddirectly from the best available copy.Available to DOE and DOE contractors fromOffice of Scientific and Technical InformationPO BOX 62Oak Ridge, TN 37831Prices available from (615) 576-8401, FTS 626-8401Available to the public fromNational Technical Information ServiceUS Department of Commerce5285 Port Royal RdSpringfield, VA 22161NTIS price codesPrinted copy A05Microfiche copy AO1


SAND92-1911Unlimited ReleasePrinted June 1993DistributionCategory UC-721Permeability and Hydraulic Diffusivity of WasteIsolation Pilot Plant Repository Salt Inferred fromSmall-Scale Brine Inflow ExperimentsD. F. McTigueThermal and Fluid Engineering DepartmentSandia National LaboratoriesAlbuquerque, New Mexico 87185ABSTRACTBrine seepage to 17 boreholes in salt at the Waste Isolation Pilot Plant (WIPP) facilityhorizon has been monitored for several years. A simple model for one-dimensional,radial, darcy flow due to relaxation of ambient pore-water pressure is applied to analyzethe field data. Fits of the model response to the data yield estimates of two parametersthat characterize the magnitude of the flow and the time scale over which it evolves.With further assumptions, these parameters are related to the permeability and thehydraulic diffusivity of the salt. For those data that are consistent with the modelprediction, estimated permeabilities are typically 10-22 to 10-21 m2. The relatively smallrange of inferred permeabilities reflects the observation that the measured seepagefluxes are fairly consistent from hole to hole, of the order of 10-10 m/s. Estimateddiffusivities are typically 10-10 to 10-8 m2/S. The greater scatter in inferred hydraulicdiffusivities is due to the difficulty of matching the idealized model history to theobserved evolution of the flows. The data obtained from several of the monitored holesare not consistent with the simple model adopted here; material properties could not beinferred in these cases.


CONTENTS1.0 Introduction . . ...0.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.0 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 32.2 Scaling of Unknown Pmameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...63.0 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.0 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1 Diffusion of Brine Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..l35.0 Radid Flow toan Open Borehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..l55.1 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..l55.2 Fluid Flux at the Borehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..l85.3 Cumulative Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..o . . . . . . . . ..2l5.4 Late-Time, Asymptotic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...235.5 Flow Limited to Discrete Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3o6.0 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3l6.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3l6.2 Fits Based on the Full Flux Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...326.2.1 Deleted Data and Other Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...336.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...356.3 Fits Based onthe Full Cumulative Volume Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...456.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...466.4 Fits Based on the Late-Time Flux Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...516.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...537.0 Sumwy and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...637.1 Choice of Parameters for the Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...667.2 Validity of the Radid Flow Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...677.3 Comparison of Fits Over Various Time Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...687.4 Comparison of Fits by Various Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..7O7.5 Additional Sources of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..7l7.5.1 Initial Brine Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..7l7.5.2 Brine Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...727.5.3 Brine Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...727.6 Expected Capacitance for an Elastic Matrix and Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...738.0 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75. . .111


FIGURES2-15-15-25-35-45-56-16-26-36-46-56-66-76-86-96-106-116-126-136-146-156-166-176-186-196-206-216-226-236-246-256-266-276-286-296-306-31Locations of the boreholes considered in this study ..............................................4Definition sketch forradid flow toaborehole .. . . .. . . .. . .. . .. . . .. . . . .. . .. .. . . .. . . . . .. .. . . . . . . ...l6Pore pressure profiles at various times based on Equation (31); radial coordinate, r,is normalized by borehole radius, u; pressure, p, is normalized by initial value pm . . . . . . . .19Fluid flux at a borehole, evaluated from “exact” solution, Equation (35) . . . . . . . . . . . . . . . . . ...25Comparison of late-time approximations (Equation41 ) and “exact” solution(Equation 35) for fluid flux at a borehole; time, t, is normalized by characteristicdiffusion time, a2/c; flux, Iqrl(a,t), isnormalizedby go =kpm/~. . . . . . . . . . . . . . . . . . . . ...26Comparison of “exact” and late-time, asymptotic solutions for inverse of borehole flux;time, t, is normalized by characteristic diffusion time, a2 / c; flux, qr (a,?), isnomdized byqo=kpW /p . .. . . .. .. .. .. . .. . . .. . . .. . .. . . .. . . .. . .. . . .. .. . . .. . .. . . . . . . .. . . . . . . . ....29Comparison of model fit todata for full flux history, hole DBTIO . .. . . . . . . . .. . . . .. . . . . . . . ...37Comparison of model fit to data for full flux history, holeDBT11 . .. . . . . . . . . . .. . .. . .. . .. . ...37Compmison ofmodel fittodata for full flux histo~, hole DBT12 . . . . .. . . . . . . . . . .. . .. . . . . ...38Comparison of model fittodata for full flux history, hole DBT13 . . . . . . . .. . .. . . . . . . . . . . . . ...38Comparison of model fit to data for full flux history, hole DBT14A . . . .. . .. . . . . . .. . . . . . . . . . .39Comparison of model fit to data for full flux history, hole DBT14B .. . . . . .. . .. . . .. . .. . . . . ...39Comparison of model fit to data for full flux history, holeDBT15A .. . .. . .. . . . . . . . . . . . .. . .. .40Comparison of model fit to data for full flux history, hole DBT15B . . . . . . .. . .. . . . . . . . . . . . . . .40Comparison of model fit to data for full flux history, hole DBT31A . . . . . . . . . . . . . . . . . . . . . . ...41Comparison of model fit todata for full flux history, hole DBT31C .. . .. . . . . . . . . .. . . . . .. . ...41Comparison of model fit to data for full flux history, hole DBT32C .. . .. . . . . . . . . .. . .. . . . . .. .42Compwison ofmodel fittodata for full flux histo~, hole UBOl . . . .. . . . . . . . . . . . . . . . . . . . . ...42Comparison of model fit to data for full flux history, hole QPBO1 . .. . . . . . . . . . . . . . . . . . . . . . . . .43Comparison of model fit to data for full flux history, hole QPB02 . . . . .. . . . . . . . . . . . . . . . . . . . . .43Comparison of model fit to data for full flux history, hole QPB03 . . . . .. . . . . . . . . . . . . . . . . . . . . .44Comparison of model fit to data for full flux history, Room D holes grouped . . . . . . . . . . . ...44Comparison of model fitto data forfullfluxhistory,Room Q Access Driftholes grouped .. . .. . . . .. . . .. . .. . . .. .. . .. . ... . .. .. . .. .. .. .. . .. . . .. . .. . . .. . .. . . . . .. . . .. . . .. .. . . . .. .. ....45Comparison of model fit to data for cumulative volume history, hole DBTIO ...............47Comparison of model fit to data for cumulative volume history, hole DBT 11...............47Comparison of model fit to data for cumulative volume history, hole DBT 12...............48Comparison of model fit to data for cumulative volume history, hole DBT13 ...............48Comparison of model fit to data for cumulative volume history, hole DBT 14A .............49Comparison of model fit to data for cumulative volume history, hole DBT 15...............49Comparison of model fit to data for cumulative volume history, hole L4B01 . . . . . . .. . . . . ...50Comparison of model fit to data for cumulative volume history, hole QPBO 1...............50Comparison of model fit to data for cumulative volume history, hole QPB02 ...............5 1Comparison of model fit to data for late-time flux, hole DBT 10. Points included inlinearegression areshaded .. . . .. . . .. .. . .. . . .. . .. . . .. . .. . . .. . . .. . . .. . . . .. . . .. . .. .. . . . . . .. . . . . . . . ...54Comparison of model fit to data for late-time flux, hole DBT 11. Points included inlinear regression are shaded . . .. . . .. .. ... . . .. . . .. . .. . . .. . . .. . .. . . .. . .. . . . . .. . . .. . . . . . .. .. . .. . . .. ....54Comparison of model fit to data for late-time flux, hole DBT 12. Points included inlinewregression areshaded .. . . .. . .. . ... .. . . .. . .. . . .. . . .. . .. . .. . . . .. . .. .. . . .. . .. .. . . . . .. . . . . . . . . ...55Comparison of model fit to data for late-time flux, hole DBT 13. Points included inlinearegression areshaded . . .. . . .. . ... . .. .. . . .. .. . . .. . .. . .. . . . . .. . .. . . .. . . . . .. . .. .. . . . .. .. . . . . . ...55Comparison of model fit to data for late-time flux, hole DBT14A. Points included inlinemregression areshaded .. . .. . .. . . .. . . .. . .. .. . .. . . .. . . .. . .. . . .. . . . .. . . .. . . . .. . . . . .. . . . . . . . .. ....56iv


1.0 INTRODUCTIONMeasurable accumulations of brine have been observed in some drillholes and excavations atthe Waste Isolation Pilot Plant (WIPP) facility horizon for a number of years. Because thepresence of brine may affect backfill consolidation, gas generation, room closure, and theperformance of the seals and waste package, a predictive model is desired. A successful modelwould allow calculations of the brine inflow rate, or, equivalently, the cumulative volume of brineas it evolves through time for a given room geometry. Any model that purports to represent theessential brine transport process will require as input numerical values of various materialproperties. The purpose of this report is to document a first attempt to infer such parameters fromfield data collected over a period of several years.The study invokes one of the simplest models that might be assumed to represent the flow ofbrine through salt. The salt is represented as a fluid-saturated, porous medium in which the flux ofbrine is assumed to be governed by Darcy’s law. Classical arguments lead to a description of brinetransport in the form of a linear diffusion problem. This parallels conventional problems ofhydrological flows in confined aquifers. An exact solution is known for the transient, onedimensional(radial) flow to a long borehole in an unbounded domain initially at uniform pressure.The behavior of the solution depends on two groups of parameters. Fits of the analytical functionto the field data thus yield estimates of two unknown parameters that characterize the system. Inthe present case, three unknown parameters arise in the analysis: the initial pressure of the brine,the permeability of the salt, and the hydraulic diffusivity of the brine in the salt. (Variouscombinations of these parameters are equally valid choices of constants to be determined by fittingthe data.) Independent measurements have indicated typical values for the initial brine pressure inthe salt, which can be used to isolate values for permeability and diffusivity.The field data used to infer the hydraulic properties of the salt are from the small-scale brineinflowexperiments conducted in the WIPP facility beginning in September 1987. The datacollectionprogram is detailed in a report by Finley et al. (1992), which presents and discusses datacollected through early June 1991. The analysis reported here considers only the data obtainedthrough mid-January 1990; later data were not available when this analysis was performed. Theexperiments consist of regular collection and weighing of the brine (if any) accumulated in each of17 boreholes. This provides a measure of the integrated brine flux over the collection interval. Ameasure of the average flow rate can also be obtained simply by dividing the volume collected bythe sampling interval. The observed cumulative volume and flux histories provide the data to1


which the model is fit. The fits seek the set of unknown parameters that minimizes the differencebetween the observationsand the model calculations.The model on which the interpretation of these data is based is highly idealized: onedimensional,radial, darcy flow, assuming a homogeneous, isotropic medium and a uniform initialbrine pressure. In this context, the model can be fit meaningfully only to “well-behaved” data(i.e., holes that exhibit a relatively high initial flux and a smooth, monotonic decay). Nonetheless,it is worthwhile to see the extent to which the salt appears to respond as a classical hydrologicalsystem. Furthermore, in those cases where the model appears to yield a reasonable approximationto the observed behavior, the fitting exercise yields estimates of critical material properties.The exercise also reveals some limitations of the model. While many of the monitored holesshow behavior consistent with the model, some do not. Evidently, more complex processes notconsidered here influence the yield of brine in some holes. Even in the context of the classicalmodel, such phenomena as heterogeneity and anisotropy in the material properties, nonuniforminitial pressure, and the multi-dimensionality of the true configuration of the monitored holes mayexert significant influence. In addition, intersection of the holes with discrete fractures wouldintroduce different boundary conditions for which the present analysis does not apply. Beyondthese complications are the possibilities of more complex interactions between the deformation ofthe salt and the brine flow, as well as multiphase flow (e.g., exsolution of dissolved gases,imbibition of room air, seepage of brine through unsaturated salt, etc.).Section 2 of this report summarizes the data collection methods and general observations aboutthe brine inflow. The data reduction scheme is described in Section 3. Section 4 provides thebackground of the model that is assumed to represent the transport of brine in salt. The solutionfor flow to a borehole and the numerical evaluation method are described in Section 5. Theparameter-estimation scheme and the results for various model approaches are described in Section6. Finally, the results are summarized and discussed in Section 7.


2.0 DATA COLLECTION2.1 General ObservationsThis report records preliminary reduction of data from boreholes drilled in salt in the WIPPfacility horizon and provides a brief discussion of their interpretation in view of a classical darcyflowmodel. Brine flow to 17 boreholes in Room D, Room L4, and the Room Q Access Drift hasbeen monitored since January 1987, May 1989, and April 1989, respectively. This report treatsonly data collected until January 1990, covering a period of up to 850 days. A detailed descriptionof the data collection process and the results through June 1991 is given by Finley, et al. (1992).The stratigraphy penetrated by the 17 holes is shown in Figure 1-1. Their dimensions andhistories are summarized in Table 1-1. Holes that were extended in length or enlarged in radius aredesignated here by the letters A, B, and C for each successive stage. The hole locations in eachroom are shown in Finley et al. (1992).Although there is considerable variability in the observed response of the holes, there aresignificant similarities. Except for the two inclined holes DBT16 and DBT17, all monitored holesyielded some brine, with maximum total mass flow rates of 2 to 25 g/d during the periodconsidered in this report. The singleexception was hole QPB02, which yielded over 100 g/d forthe first 100 days.In many cases, the flow rate declined in a Fairly smooth fashion over time, which is theexpected behavior of an open borehole in a classical hydrological system (see Section 5.2). HolesDBT 10 to DBT 15 showed this response at early time, although the flow to DBT 14 actuallyincreased with time after about 100 days. All these holes showed somewhat more erratic behaviorafter several hundred days, with occasional increases in flow rate (DBT 12) or an apparent levelingoffof the flow (DBT 13). Holes DBT31 and DBT32 were initially 4-inch-diameter (radius a =0.051 m) holes; DBT31 showed a decline in flow at early time, while DBT32 was erratic. Afterextension and then enlargement, these holes exhibited erratic responses again, although tentativeinformation was extracted from the period of declining flow in DBT32 at about 450 days.3


UnitClay Grn1- 0 1- 04 m? 1=1=1= 1=m m m m mn n n c1 nClay F[u cL4X01L4B01 IClay1! 11I 1iPH-3~xxxx1ElN•1 xxxEl———LegendHaliteAnhydriePolyhalitic HaliteArgillaceous HaliteClay SeamWasteFacilityHorizon,Vertical Scale(No Horizontal Scale)8 ft62m+411200TRI-6344-553-3Figure 2-1. Locations of the boreholes considered in this study (from Finley, et al., 1992).4


Table 2-1. Borehole CharacteristicsLength RadiusHole Operation Date (m) (m)DBTIO drilled 9/1 8187 5.334 0.051DBT11 drilled 9/23/87 4.633 0.051DBT12 drilled 9122/87 3.688 0.051DBT13 drilled 9/1 ‘718’7 2.804 0.051DBT14A drilled 9/14/87 2.591 0.051DBT14B extended 6/30/88 5.608 0.051DBT15A drilled 9/1 5/87 2.743 0.051DBT15B extended 7/5/88 5.791 0.051DBTI 6A drilled 9/25/87 2.540 0.051DBT 16B extended 7/27/88 5.169 0.051DBT17A drilled 9/29/87 2.540 0.051DBT 17B extended 7/28/88 5.436 0.051DBT31 A drilled 9/8/87 2.134 0.051DBT31B extended 215/88 4.877 0.051DBT31 c enlarged 5/1 8/88 5.639 0.457DBT32A drilled 9/1 0/87 2.896 0.051DBT32B extended 2/9/88 5.664 0.051DBT32C enlarged 5/24/88 5.664 0.457L4B01 drilled 5/ 16/89 5.791 0.051L4XOI drilled 5/4/89 5.715 0.457QPBO1 drilled 4/ 18/89 3.048 0.025QPB02 drilled 4/1 8/89 3.099 0.025QPB03 drilled 4/ 18/89 3.099 0.025QPB04 drilled 5/1/89 3.073 0.025QPB05 drilled 5/1/89 3.099 0.025Hole L4B01 (radius 0.051 m) exhibited a smooth decay in flux that allows a fit of the simplemodel. The nearby hole L4X01, situated in the same stratigraphic horizon and of the same lengthand angle, is of much greater radius, 0.46 m. Hole L4XOI showed a high initial flux andsubsequent decay, but the decay was quite rapid, with the flux reaching zero at about 170 days,5


Because the observed flux-decay rate to L4X01 does not decrease continuously, a fit of the modelcould not be obtained.Among the boreholes in the Room Q access drift, hole QPBO 1 shows a response over theconsidered time interval that is most consistent with the idealized model and allows a fit todetermine material parameters. As noted above, hole QPB02 yielded a much greater quantity ofbrine than any other hole monitored in this study. It showed a relatively smooth decay in flux overthe first 46 days, followed by a steeper decline until about 200 days, when the flux was at leasttwo orders of magnitude smaller than that calculated from the earliest observation. A discretefracture with a detectable offset was identified by a borehole camera in QPB02 in March 1991(Finley et al., 1992); this provides one possible explanation for the anomalously large flux. HoleQPB03 showed relatively small changes in flux over time, but a fit of the model was obtained.Holes QPB04 and QPB05 yielded no brine for the first several samplings, and they exhibitedrather erratic accumulations.No fits could be obtained for these data.2.2 Scaling of Unknown ParametersSimple scaling arguments based on the darcy flow model, which are developed in more detailin Sections 4 and 5, indicate the orders of magnitude of the parameters of interest. Thepermeability is shown to scale like qO~a/ pm, where qO is the fluid flux, p is the fluid viscosity, ais the borehole radius, and pm is the initial pressure. Typical fluxes, based on the total volumeflow rates divided by borehole wall areas, are of the order of qO- 10-’0 m/s; the brine viscosity isof the order of p. - 10-3 P&s; initial pressures are believed to be of the order of pm - 107 Pa. Thescaling argument thus indicates permeabilitics of the order of 10-21 m2, and this is borne out by themore elaborate fitting exercise.The model also shows that the fluid diffusivity, c, scales like c - az /tO, where tO is acharacteristic time, defined as the time at which the flow has fallen off to e-l times the observedmaximum. For those holes that exhibited a smooth decline in flow rate, totypically ranges fromseveral tens to several hundreds of days, which implies a hydraulic diffusivity of the order of 10-10to 10-9 mz/s. This is again borne out by many of the detailed fits, but the diffusivity is ratherdifficult to determine with confidence from these data.6


3.0 DATA REDUCTIONData were collected for the mass of brine accumulated over a given period between samples.Volumes were computed by dividing the mass by the brine density, taken hereto be 1200 kg/mq.Volumes per unit area of borehole wall were calculated by dividing the volume by the cylindricalsurface area of each hole (the end area is neglected), 2naL, where L is the length of the hole.Fluxes were calculated by a “centered difference” approximation (i.e., volume per unit area dividedby time between samplings with the resulting flux assigned to the midpoint time of the interval).Note that Finley et al. ( 1992) report mass-flow rates based on a “backward difference”approximation (i.e., mass divided by time between sampling with the resulting mass-flow rateassigned to the time of the measurement).The foregoing procedure averages the total brine seepage over the entire borehole wall area.The salt intersected by the hole is assumed to be homogeneous, and the seepage is assumed to beuniform along the length of the hole. Thus, parameter estimation based on data treated in thismanner yields efiective values of the parameters over a sampling scale of the order of the boreholelength. In general, the field sampling method cannot discriminate between uniformly distributedflow and localized seepage in discrete horizons intersected by a particular hole. However, in caseswhere there is reason to believe that the flow is derived principally from some fraction of theborehole length, the appropriate corrections to the parameter estimates are straightforward. Thisprocedure is discussed further in Section 5.5.7


4.0 MODEL DESCRIPTION4.1 Diffusion of Brine PressureThe model that underlies much of classical hydrology results in a linear diffusion equation forthe excess pore pressure or, equivalently, hydraulic head in the fluid. Both standard texts and theresearch literature discuss many different ways of deriving this governing equation (e.g., Bear,1972). One such derivation is summarized briefly here for completeness, and for reference in thesubsequent discussion of results. This particular development is equivalent to, or contains asspecial cases, most of the classical models.Conservation of mass for the fluid component of a saturated, porous medium is given by~+vat(pvf)=o,(1)where p is the partial density of the fluid, defined as mass of fluid per unit volume of thefluid/solid system, and v~is the fluid velocity, averaged on a suitable scale to represent the volumeflux of fluid per unit area of fluid. It is convenient to decompose the partial density into theproduct of the porosity, & and the material density, y} defined as mass of fluid per unit volume offluid:(2)Substitution of (2) into (1), rearrangement of terms, and linearization lead to the followingstatement of the fluid mass balance:(3)where subscript zeroes indicate constant reference values of the parameters, v~ is the solid skeletonvelocity (again averaged on an appropriate scale) and q is the darcy flux, or “seepage velocity, ”defined in terms of the relative velocity of the fluid and solid, q = @,,(vf – v,,). A similar statementof mass conservationfor the solid skeleton takes the form:


a@+(Ho)a—— J+(l–$,,)V.V, =O,at-)’.,0 at(4)where y~ is the material density of the solid.Equations (3) and (4) sum to give:—— @o%+(HOPYYf!l at -1’s” a~+v”q+v”v’’=o”(5)For incompressible constituents, yf = yfO and y$ = y$O are constants, the first two terms vanish,and Equation (5) simply states that an influx of fluid into a volume element of the porous mediumis balanced by dilatation of the solid skeleton, or conversely: an efflux of fluid is balanced bycompaction. The more general case given by Equation (5), allowing for compressible constituents,is treated in the remainder of this section.To complete the model, it is necessary to stipulate constitutive equations that represent thebehavior of the materials of interest. In particular, the dependence of yf, y$, q, and V v,, on thefluid pore pressure p must be given. First, the fluid density is assumed to depend linearly on thepore pressure:‘f=yfo[’++f(p-pol(6)where Kf is the fluid bulk modulus (the inverse of the fluid compressibility), and p. is the fluidpressure in the reference state. Second, the solid density is assumed to depend on the mean totalnormal stress, O, and the fluid pressure:“$=‘sO‘-(1{-$:)K, [(0-%)}+$O(P-PO)] ~(7)where KS is the bulk modulus of the solid, and CJOis the mean total stress in the reference state.Compression is negative in the sign convention adopted here. The mean stress, o, is defined asone-third the trace of the total stress. The mean stress is thus minus the “confining pressure” oftraditional usage in soil and rock mechanics. The fluid flux is assumed to follow Darcy’s law,which is a statement of the quasi-static balance of momentum for the fluid:10


q = +’(P–-rfow) , (8)where k is the permeability (here assumed to be isotropic), p is the fluid viscosity, g is themagnitude of the gravitational acceleration, and the z coordinate is vertical and positive upward.Finally, the dilatation rate, V. v,, is identified (for small deformation) with the rate of change ofthe volumetric strain, e, which, in turn, is related to the mean stress and fluid pressure viaHooke’s Law for a linearly elastic material:‘v:*=*$[’o-oO’+[l(9)where K is the bulk modulus of the “drained” (p – p. = O) porous skeleton.Substitution of Equations (6) through (9) into Equation (5), along withuniform permeability, k, and viscosity, y, givesthe assumptionofwhereakatC’~––vV2p=–B’~,(10)and“=H+2[1-3B’=+ l–f .[) S(11)(12)The compatibilityconstraint for a porous, elastic material can be written in the formv2~ = –2(1 – 2V) ~BJV2p .3(1- v)(13)Equations(10) and (13) can be combined in the formc—––v2p=–% : (%-’v’”)(14)11


where C is the capacitance,~ = ~, ~_ 2(1 – 2v) KB’21[ 3(1- v) c’ ‘(15)and B is a source coefficient:B = B, ~_ 2(1 – 2v) KB’Z1[3(1 – v) c’ ‘(16)The parameter con the right side of Equation (14) is the hydraulic diffusivity,defined byk~=—pc “(17)The quantity B\C is known as Skempton’s coefficient and represents the ratio of the pore-pressureincrement to the magnitude of the mean stress change under undrained conditions (Rice andCleary, 1976).Equations (7), (9), and (13) introduce the mean total stress, (s= tr cJ/ 3, as an additional fieldvariable that must be determined as part of any application of the model. In general, this requiressolution of the balance of linear momentum, or the “equilibrium” equation of classical elastostatics,V G + pog = O, Where p. = yfO(l – $.) + y~O@Ois the total density, and g is the gravitationalacceleration vector. In some special cases, such as that of radial flow in an unbounded domain, themean stress change can be shown to be identically zero (Nowak and McTigue, 1987).Furthermore, it is often appropriate in more general applications to approximate the mean stress asremaining constant. This stipulation reduces Equation (14) to a linear diffusion equation, or a“heat equation,” for the pore pressure alone:(3 LLV2P=0.at p(18)Equation ( 18) is fundamental to classical hydrology, although it often appears in terms of headrather than pressure. The identity is made by defining the head h = p/ Yj,g, so that Equation (18)becomes(19)where S = yjOgC is the specific storativity, and K = ~JOg / p is the hydraulic conductivity.12


4.2 AnisotropyThe flow to a circular borchole of finite length and bounded on one end by a plane surface is atwo-dimensional problem, even when axisymmetry can be supposed. However, if thepermeability is strongly anisotropic, which is often the case in geological materials, the flow can bepredominantly in planes normal to the axis of a vertical borehole.The flow to an open borehole is idealized here as a one-dimensional process (i.e., radial flowoccurs in planes normal to the borehole axis). In this configuration, the assumption of isotropy ofthe hydraulic properties can be viewed as the somewhat less restrictive assumption of transverseisotropy. For example, note that no flow was observed in the two sub-horizontal holes drilledfrom Room D, while eight vertical holes in the same area produced brine. This might beinterpreted as an indication of flow confined predominantly to horizontal planes, perhapscorresponding to compositional or mechanical layering in the rock. This view, however, is notconsistent with the observed accumulation of brine in the two sub-horizontal holes drilled in RoomL. The sub-horizontal holes from Room D were drilled in relatively pure halite, while those fromRoom L were drilled in argillaceous halite (Finley et al., 1992).13


5.0 RADIAL FLOW TO AN OPEN BOREHOLE5.1 Pressure FieldThe radial flow model assumes that an open, circular hole is introduced into an unbounded,homogeneous domain at time t = O. A definition sketch of the model is shown in Figure 5-1. Theinitial fluid pressure, pm, is assumed to be uniform. The open face of the borehole is atatmospheric pressure, p = O, causing flow toward the hole, associated with relaxation of thepressure in a zone that grows diffusively outward from the hole. The exact statement of the initialvalueproblem is, from Equation (18):——ap——Ca ag .0d? rib () ih- ‘(20)with initial condition:p(r,o) = pm , (21)and boundary conditions:p(a,t)= 0, (22)lim p(r,t) = pm , (23)r+ wwhere c = k / pC is the hydraulic diffusivity and a is the radius of the borehole. An analyticalsolution to Equations(20) through (23) is well known:L=_?Pcajoew(-~’t.)f(~;~x)z,u(24)wheref(u; r.)=.lo(urx)~(u) – ~(ur*)Jo(u)J;(u) + Y02(U) ‘(25)and where tx= ct / a’, rx = r/a,,order zero, respectively (Crank, 1975).and Jo and Y. are Bessel functions of the first and second kind of15


p(a,t)=O\/p(r, t) =POa00TRI-61 19-276-0Figure 5-1. Definition sketch for radial flow to a borehole.16


Although Equation (24) is a closed-form solution to the problem, it is difficult to evaluateaccurately because the integrand is singular at the lower limit of integration. However, thesingularity is integrable, so that Equation (24) can be evaluated accurately if appropriate care istaken.To isolate the singular part of the integrand, one can partition the integral in Equationtwo parts:(24) into(26)where & can be chosen to be arbitrarily small. The expansions of JO(L) and YO(&)for smallargument are given by:JOK)=I-;+..., (27)(28)where ~= exp (y), and y = 0.5772 ... is Euler’s constant. Substitution of Equations (27) and (28)into the first integral in Equation (26), and expansion about u = O gives, to leading order:cJolim– z exp(–u2t.) ~(u, ~)~ = in r. C exp(–u2t. ) —+... du.14+11 x u J ln2(~u/2) un(29)The right-handside of Equation (29) can be integrated by parts to yield(30)Thus, Equation (24) can be evaluated accurately by separating the singular part of the integrand,integrating it analytically, and evaluating the remainder by numerical quadrature. Substitution ofEquation (30) into Equation (26) gives the final form used:P—. —Pm–exp(–&2t. ) in K 2 -ln(@/2)-;j’ exP(-@f*Mu;~*)* -c(31)


Evaluations of Equation (31) have been carried out, setting &= 10-8, and performing theintegral by the Gauss quadrature routine DGAUS8 in the SLATEC subroutine package. The errortolerance in the numerical integrator was set to ERR = 10-6, and the calculations were carried outon a VAX 8650 using double-precision arithmetic. Figure 5-2 shows Equation (3 1) evaluated for1< r. S 5 and t. = 0.01, 0.1, 0.5, 1.0, 5.0, 10.0, 50.0. The results shown here differ slightlyfrom those reported by Nowak and McTigue (1986) and Nowak, McTigue, and Beraun (1988)because of the more careful treatment of the integral in Equation (24) near u=O. The mostsignificant differences are evident at later times and larger radial coordinates, where the pressuresare typically somewhat larger than in the original calculations. Thus, the numerical quadratureapplied directly to Equation (24) appears to underestimate the contribution near the singularity.The exact numerical values on which Figure 5-2 is based are reproduced in Table 5-1 forreference. Note that the integration scheme appears to have difficulty resolving valuesasymptotically close to unity. For example, at t* = 0.01, the dimensionless pressure reaches amaximum value of 0.9986 at r* = 1.6, and decreases monotonically to 0.9952 at r* = 5.0, ratherthan continuing to approach 1.0. No attempt was made to resolve this problem for the calculationsshown in Table 5-1, as it appears to have little practical consequence. However, for reference, theasymptotic expansion of Equation (24) for large X = (r* – 1)/2A which can be used to obtain amore accurate evaluation of the pressure far from the hole, is given by:lim ~=1– ‘X-l exp(–X2)+... ,x+- p. J=(32)5.2 Fluid Flux at the BoreholeA quantity that can be estimatedobtained from Equation (24) by application of Darcy’s law:from field data is the flux into the hole, q,(a, t), which can be(33)where k is the permeability and p is the brine viscosity. The resulting expression has been notedpreviously by Nowak and McTigue ( 1987) and Nowak et al. (1988):(34)


0 Qa1.00.80.60.4c1 t = 0.01A t= 0.10+ t = 0.50x t= 1.000.20 t = 5.00v t=lo.oom t = 50.000.01.0 2.0 3.0 4.0 5.0rlaTRI.61 19.277.0Figure 5-2. Pore pressure profiles at various times based on Equation (31); radial coordinate, r,is normalized by borehole radius, a; pressure, p, is normalized by initial value pm.19


Table 5-1. Pressure Profiles at Various Times Based on Equation (31)r*t* = 0.01?*=O.1t* = 0.5t*= 1.0t* = 5.0t* = 10.0t* = 50.01.00.00000.00000.00000.00000.00000.00000.00001.20.85570.39870.22340.17850.11390.09680.07021.40.99500.68300.40670.32730.21010.17850.12961.60.99860.85530.55580.45250.29300.24920.18101.80.99820.94280.67490.55800.36560.31130.22632.00.99790.97980.76790.64660.42980.36660.26682.20.99770.99270.83840.72030.48710.41640.30352.40.99740.99620.89040.78120.53860.46140.33682.60.99720.99690.92740.83080.58500.50250.36752.80.99690.99690.95290.87060.62700.54010.39593.00.99670.99670.96980.90220.66500.57470.42223.20.99650.99650.98060.92690.69940.60660.44683.40.99630.99630.98720.94580.73060.63600.46983.60.99620.99610.99120.96000.75900.66330.49143.80.99600.99600.99340.97060.78470.68860.51184.00.99590.99580.99450.97830.80790.71210.53114.20.99570.99570.99510.98380.82890.73400.54944.40.99560.99550.99530.98760.84790.75420.56684.60.99540.99540.99530.99020.86500.77310.58334.80.99530.99530.99530.99190.88040.79060.59905.00.99520.99510.99520.99310.89420.80690.6140*Radial coordinate, r, is normalized by borehole radius, a; time, t, is normalizedby a2 / c; pressure, p, is normalizedby initial value, pm.20


where q* is the magnitude of the normalized flux at the borehole, q* = qr (a, t)/ qO = –qr(a, t)/ q.,and the scale of the flux, qO, is given by qO= kpm /jM.Equation (34) encounters the same difficulty discussed above in the context of the pressureprofiles: the integrand is singular at u = O, and numerical quadrature routines cannot easily resolvethis. However, the singularity is of the same form as that in Equation (24), and is thereforeintegrable. The identical procedure can be applied to Equation (34). However, the same result canbe achieved by differentiating Equation (3 1) directly, giving:exp(–&2L) + ~ - exp(–u2L) A‘*: - ln(~&/2) nz J, J:(u)+ q’(u) ‘ u “(35)Evaluations of Equation (35) have been carried out in the same fashion as those of Equation(31 ) described above, again taking&= IO-8, and ERR= 10-6. The results are shown in Table 5-2and in Figure 5-3. The fluxes computed here are slightly higher than those obtained by Nowakand McTigue ( 1986) and Nowak, McTigue, and Beraun (1988), again because numericalquadrature applied directly to Equation (34) underestimates the contribution near the singularity.A check against independent calculations is possible for t* 0.99.5.3 Cumulative VolumeThe cumulative volume of fluid per unit area of borehole wall, v, is obtained by integratingEquation (35) over time:v.~–t.4 m[1-exp(-u2t.)]@+ ~ &2t~ln(~&/2) + Z J J:(u) + Y;(u) us H ln& ‘c(36)where w = V/V.,and V[J= kpma / ~c = Cpma is the reference volume scale.21


Table 5-2. Fluid flux at a borehole*t+ q*Equation (35) J&C ( 1942)O.lE-01 6.1291 6.1290.2E-01 4.4718 4.4720.4E-01 3.2969 3.2970.6E-01 2.7748 2.7750.8E-01 2.4625 2.4620. lE+OO 2.2489 2.2490.2E+O0 1.7154 1.7150.4E+O0 1.3326 1.3330.6E+O0 1.1601 1.1600.8E+O0 1.0559 1.056O.l E+O1 0.98390.2E+01 0.80070.4E+01 0.66450.6E+01 0.60100.8E+01 0.56170.1E+02 0.53400.2E+02 0.46130.4E+02 0.40410.6E+02 0.37620.8E+02 0.35840.1 E+03 0.34570.2E+03 0.31090.4E+03 0.28220.6E+03 0.26760.8E+03 0.25800.1 E+04 0.25110.2E+04 0.23160.4E+04 0.21490.6E+04 0.20610.8E+04 0.20030.1 E+05 0.1961*J&C = Jaeger and Clarke22


5.4 Late-Time, Asymptotic FluxThe asymptotic expansion of Equation (34) for t.>>1 takes a particularly simple form usefulfor fitting data. The development is reproduced herein detail.In order to find the late-time, asymptotic expansion of the fluid flux (Equation 34), letqz = uztx, so that large ?*corresponds to small arguments of the Bessel functions. Introduce theappropriate expansions for small argument from Equations (27) and (28), and expand theintegrand:(37)where ~ = exp(y), and y = 0.5772 . . . is Euler’s constant. Integration of each term in Equation(37) by parts yields:)11-J q(q’ -1) f=p(-mz)q exp(-~’) ~lim q. = –2 -~(T4-3q2+1 +... dqf.+- r 0 In (~q/2tl’2) [—— dq+”..t. o in (flq/2t~’2)(38)Expansionof 1/ ln(~q/ 2tj”) in powers of 1/ ln(4t, / P2) gives:1 221n’rl 4(ln@2ln(~q/2t!’2) = - ln(4t. /~2) 1+ ln(4t. /132)+ [ln(4tx /~Z)]2 +”””[}(39)Substitution of Equation (39) into Equation (38) and integration term-by-term yields:23


2X2 I 3- ln(4L /~2) [0+0+ [ln(4t:/p2)12 +”””1 (40)21t.1/21+ ln(4t. /~2) 0+ln(4t. /~2)+ ””” “[Combination of the first four nonzero terms in Equation (40), and neglect of the smaller term oforder t~’ in–2 (t*), yields the final result:limq, =r.+-2ln(4L/~2)()?!&y21(41)1- ln(4~/~2) - [ln(4t, /~1)]2 ‘“”” “Figure 5-4 shows an evaluation of the full integral solution for the flux from Equation (35),along with the late-time approximations based on Equation (41). It is evident by inspection ofFigure 5-4 that one or two terms of the series given by Equation(41) capture the general trend ofthe exact solution at late time, but overestimate the flux somewhat. The two-term expansion yieldsa flux that is 5’-ZO too high at tx= 100 and 370 too high at tx = 1000. The three-term expansion isvery close (less than 2?10 error) for tx > 100. Table 5-3 presents the late-time approximationscomparedto the “exact” fluid-flux solution.A simple scheme for fitting data to extract hydraulic properties is suggested by taking theinverse of Equation (41):7c’16lim q.-] = ~ln ~ 1+(42)/.+-( P2 ){ 1n(4:/P2) + [ln(4t. /~z)]2 +””” I24


2.5I I I I I I Ill I I I I I I Ill I I I I I I Ill I I I I I If2.01.5$L1.00.50.0,0-1I I I I 1 I I I I I I I I I I II I 1 I I I Ill I I I I 1 I10° 10’,(32 ,03TimeTRI.61 1%278.0Figure 5-3. Fluid flux at a borehole, evaluated from “exact” solution, Equation (35).25


1.0I I I I I I I II1 I I 1 I I 1 I— Integral Solution0.80000,0.00000. l-TermLate-------” 2-Term Late.,,, ... —!— 3-Term Late... ..,..,.,,.,.,,,........~L0.6* ...,●✎☛✎ ✎‘. .,,-.0.40.20.010’ 102,03TimeTRI.61 19.279.0Figure 5-4. Comparison of late-time approximations (Equation41 ) and “exact” solution(Equation 35) for fluid flux at a borehole; time, t, is normalized by characteristicdiffusion time, a2 /c; flux, Iqrl(u,t), is normalized by go = kp~ 1P.26


Table 5-3. Comparison of Late-time Armroximations . .and “Exact” Solution for Fluid Flux at aBoreholetx(Iq*Late-time Approx.“Exact”1Term I 2 Terms I 3 Terms100.78910.60940.44830.5340500.48260.41540.37850.38831000.41350.36410.34100.34575000.31020.28250.27270.273910000.28010.25750.25030.251150000.22860.21350.20960.2100100000.21180.19890.19570.1961Retaining the first two terms in Equation (42) (i.e., neglecting the term of O(ln-’ (4t. / ~’)),andreturning to dimensionalvariables, this takes the convenient form:Iim Iqrl-](a, t)= A in t+ B, (43)t*+mwhereA– KU2kpm ‘(44)and[1B=Aln — 4C(45)~’p “Thus, a plot of the inverse flux at the borehole versus In t is approximately linear at sufficiently latetime. The slope, A, is an indicator of the permeability, and the intercept, B, is an indicator of thehydraulic diffusivity through the simple relations:k– vu2pmA ‘(46)and~.—a’~4exp ~()(two - term expansion).(47)27


Note that, if only the first term of the expansion given in Equation (42) is retained, Equationreplaced by(47) isc.—~2p24exp — ~ (leading order).()(48)Thus, the higher-order approximation simply shifts the curve upward by a constant, Ay. As aresult, a fit based on only the first term in Equation (42) will yield an estimate for the diffusivity(Equation 48) that is a factor of ~ ~ 1.78 times greater than that obtained in view of the two-termexpansion (Equation 47).Figure 5-5 shows a plot of inverse flux versus the logarithm of time for t. >10. The opensymbols are computed from evaluations of the exact, integral solution. The lines show one, two,and three terms of Equation (42). As noted above, “both the leading-order approximation and thenext, higher-order approximation are linear on this plot, with the latter simply shifted upward by aconstant. Equation (47) will give a better estimate of the hydraulic diffusivity than will Equation(48), but Equation (47) still tends to overestimate c. A nonlinear fit of the three-term series givenin Equation (42) would give a much better result, and, although more involved than a linearregression based on the first two terms, is still far easier to perform than a fit requiring numericalquadtldture on Equation (35).The inverse-flux fitting method was tested using synthetic data generated by numericalsimulations of radial flow (Webb, 1992). Although the permeabilities used in the simulations wererecovered with very good accuracy (typically within a few percent), the diffusivities estimatedusing Equation (47) were typically overestimated by factors of about 50%. This difficulty isinherent in the method because extrapolation of the late-time fit back to in t =0 introducessignificant error (Jaeger, 1958).28


6.0I I I T I f I I I I I I I I I I I III I I I II5.0)4.0x3ii3.02,01.0oExact— 1-Term Late.......... 2-Term Late------- 3-Term Latet0.0,01 102 103 104TimeTRI.61 19.280-0Figure 5-5. Comparison of “exact” and late-time, asymptotic solutions for inverse of boreholeflux; time, t,is normalized by characteristic diffusion time, az / c; flux, Iqrl(a,t), isnormalized by qO= kpm I pa.29


A good estimate of the diffusivity from late-time data is difficult to obtain if other processesaffect the flow at early time. For example, suppose that the early-time flow is delayed by thegrowth of microcracks, dilatation, and accompanying “storage” of brine. After some time, theflow may behave in the classical fashion, and the slope of the lql-’ versus in t plot would reflectthe permeability. However, the time should be “re-zeroed” to account for the delay. Without somerational basis for such a correction, the diffusivity inferred by this method is inherently unreliable ifprocesses not accounted for are significant. An example that has been studied in the context of thethermal-conductivity probe is the effect of contact resistance between the probe and the medium(Blackwell, 1954).5.5 Flow Limited to Discrete HorizonsConsistent with the analytical model discussed above, the present reduction of data assumesthat the hole penetrates a domain of isotropic, homogeneous salt. Thus, production of brine isassumed to be uniform along its length. Because the brine collection method integrates over theentire borehole length, a more detailed breakdown of the distribution of flow cannot be extracted.However, if one were to assume that all of the accumulated brine in a particular hole came fromsome discrete fraction of its total length, LP / L, where LP is the producing length and L is thetotal length, then the parameters estimated here could be resealed accordingly. In particular, boththe inferred permeability and capacitance would be multiplied by (LP / L)-l to obtain theappropriate parameters for the producing layer, with the remainder of the stratigraphy regarded astotally impermeable.30


6.0 PARAMETER ESTIMATION6.1 MethodThe fits reported here for the full-flux and cumulative-volume histories were performed withthe parameter-estimation code ESTIM (Hills, 1987). The code seeks the set of model parametersthat minimizes the sum of squares of residuals, or differences between observed and simulatedquantities. In the present case, the data used are either brine flux or cumulative brine volume,estimated as discussed in Section 3. The simulations are numerical evaluations of the exactsolutions for radial seepage to a long hole, given by Equations (35) and (36) for flux and volume,respective y.Programs that evaluate the functions describing seepage to a borehole, Equations (35) and(36), are called as subroutines by ESTIM. These function evaluations were thoroughly tested asnoted in Section 5.2. Each of the analytical functions is a two-parameter description of the flow;one parameter scales the magnitude of the flux or volume, and the other is a characteristic time overwhich the flow evolves. Thus, Equation (35) can be fit to data for brine flux by seeking themagnitude scale, kpm /pa, and the time scale, a2 / c. The brine viscosity, p, and borehole radius,a, can be determined by independent measurements with some confidence, leaving the product ofthe permeability and the initial pressure, kpm, and the diffusivity, c, as unknowns. A furtherassumption for the initial pore pressure pm isolates the permeability. The capacitance is then easilycalculated from C = k/PC.Note that, with few exceptions, the observed flow rates fall in a reasonably narrow range.Thus, the product of permeability and initial pressure is fairly well constrained in most casesconsidered here. However, the detailed history of the flux for many holes was somewhat erratic,so fits for the diffusivity are more uncertain.ESTIM employs a local least-squares method (i.e., the code seeks a local minimum in the sumof squares of residuals as a function of the unknown parameters). In general, experience with thepresent data suggests that the minimum local to a reasonable initial estimate of the parameters is thedesired global minimum. ESTIM requires as input a set of initial estimates for the parameters to bedetermined, as well as upper and lower bounds on their possible values. At each iteration, thecode constructs a sensitivity matrix consisting of the partial derivatives of the error measure (thesum of squares of residuals) with respect to the unknown parameters. These derivatives were31


approximated for all calculations reported here by first-order differences. ESTIM then employs aquadratic least-squaresmethod to optimize the pammeter estimates.The parameter estimation code includes an option to perform a propagation of varianceanalysis. This option was exercised for the fits to the full flux histories reported here; it was notused for the cumulative-volume fits. The analysis provides a measure of the sensitivity of theestimated parameters to small random errors in the data. Note that the standard deviations returnedby ESTIM are “... estimates of the true standard deviations of the estimated parameters [iq1.ESTIM converges to the global minimum.2.The simulatormodels the physics of the problem well.3.4.The property models are appropriate for the materials in question.The measurements errors are random, independent, [and] have zero mean, with either auniform standard deviation ... or known and specified standard deviation” (Hills, 1987),For the propagation of variance performed on the analysis of the full flux histories, themeasurement error is assumed uniform for every sampling interval in each borehole among theentire group. In particular, the standard deviation of the measurements of mass of brine collectedwas assumed to be i5 g for each sample taken. For a uniform, seven-day sample interval, thiscorresponds to a measurement errorofiO.71 g/d on the total mass flow rate, which is comparableto the error estimates reported by Finley et al. (1992), based on laboratory repeatability tests and ona detailed statistical analysis of the field data by Rutherford (1992). This estimate for the error inthe measured mass flow rate was then divided by the brine density and the borehole wall area foreach hole to yield an estimate of the error in the flux. ESTIM returns normalized standarddeviations for each parameter, which are then multiplied by the corresponding error estimate for theraw data to yield the standard deviation associated with each parameter in a given fit. These valuesare reported in Section 6.2.2.6.2 Fits Based on the Full Flux HistoriesThe principal emphasis in this report is on fits based on the full history of the brine flux foreach borehole. Fits to the flux are believed to yield the most significant information with regard tothe hydraulic diffusivity. The least-squares fitting routine tends to place greater weight on thelarger numerical values in a given data set. Because the flux is maximum at early time, fits to the32


flux data tend to weight the early-time response. The flow changes most rapidly at early time, sothat the most important information with regard to the evolution of the flow through time, which ischamcterized by the diffusivity, is contained in the early-time data.One disadvantage of fitting the flux is that the data reduction entails an approximation to obtainflow rates. In particular, the field sampling program yields the cumulative volume over thesampling interval. In the present case, the mean flow rate over the sampling interval (an exactquantity) was calculated and assigned to a time corresponding to the midpoint of the interval (anapproximation). This approximation is valid when the rate of change of flow rate as well as thesampling interval are relatively small. A second disadvantage to fitting the flux is that the dataoften exhibit considerable scatter (i.e., while the cumulative volume must always be a nondecreasingsequence of data and often is reasonably smooth, the discrete approximations to itsderivative can be highly variable). The least-squares routine can encounter difficulty fitting datawith large scatter.6.2.1 Deleted Data and Other Data ReductionIt is often necessary to discard extreme outliers in order to obtain a good fit by least-squaresmethods.An effort was made in this study to delete a minimum of data.For many of the drillholes, no brine was obtained in the first sampling, yielding an estimatedflux of zero at the earliest available time. This datum was discarded in every case in which it arose.Because the model (Equation 34) predicts large flux at early time (q+ co as t + O), the residualbetween a computed value and a zero datum is large, and the zero is given great weight. Thus, theoptimization scheme seeks to make the diffusivity very large, and does not converge.Deleted data, as well as other data reduction applied in special cases, are summarized below forthe fits to the full flux histories; data are listed as pairs in the format (time, flux), with time inseconds and flux in meters per second:● DBTIO — deleted initial zero (2.56x 10s, 0.0) and one outlier (4.49 x 107,7.88 x 10-11,● DBT11 — deleted initial zero (4.01 x I@, 0.0)● DBT12 — deleted initial zero (8.33 x 104, 0.0)●DBT 13 — no data deleted“ DBT14A — deleted all data for increasing flux, t > 1.26x 107 s (-146 days)33


●●●●●●●●●●●●●●●●DBT14B — deleted initial zero (3.03 x 105, 0.0) and all data for increasing flux,t>1.54 x 107 s (-178 days); all flow for this time interval was assumed to be to the newlyextended section of the borehole (L= -3.0 m)DBT15A — no data deleted (through t= 2.44 x 107s ~ 285 d)DBT15B — deleted first three data (increasing flux): (3.04 x 10s, 3.02 x 10-12),(9.06 x 105, 1.37 x 10-]0), (1.51 x 106, 1.69 x 1O-10); volume due to flow to originalborehole was calculated from the fit to DBT 15A and subtracted from the raw data forDBT15B; remaining volume was assumed to be due to flow to the newly extended sectionof borehole (L= -3.0 m)DBT31A — deleted first datum (increasing flux): (6.88 x 105, 1.06 x 10-lO); deleted alldata for erratic flow t > 1.32x 107 s (-152 days)DBT31 B — no fit obtained; increasing flow rateDBT3 lC — fit only to period of declining flux from 196 to 335 days (referenced to time ofenlargement at 253 days from initial drilling for DBT31 A); deleted 27 zero-flux values fromtime of enlargement to time of first measured brine accumulation; deleted all data fort>2.87 x 107s (-332 days) when flow rate began to increaseDBT32A — no fit obtained; increasing flow rateDBT32B — no fit obtained; increasing flow rateDBT32C — fit only to period of declining flux from 203 to 322 days (referenced to time ofenlargement at 254 days from initial drilling for DBT32A); deleted 27 zero-flux values fromtime of enlargement to time of first measured brine accumulation; deleted all data fort >2.76 x 107 s (-319 days) when flow rate began to increaseL4B01 — no data deletedL4XOI — no fit obtained; data show steep, nearly linear decline to zero flux; could not befit by model that predicts continuously decreasing flow rateQPBO1 — deleted three outliers: (1.26 x 106, 1.02 XIO-l 1), (1.82 x 106,7.41 x 10-11),and (1.43 x 107, 2.38 x 10-lO); deleted all data for t >3.32 x 107 s (-384 days) when flowrate began to increaseQPB02 — deleted all data fort> 4.01 x 106 s (-46 days) when flow rate began to decreaserapidly, reached zero, and subsequently increasedQPB03 — deleted three initial zeroes: (7.72 x 104, O), (1 .97 x 10s, O), and (3.77 x 105,O); deleted one additional early-time zero: (1.4 1 x 106, O); deleted all data fort> 1.91 x107 s (-221 days) when flow rate began to increaseQPB04 — no fit obtained; no discernible period of continuously decreasing fluxQPB05 — no fit obtained; no discernible period of continuously decreasing flux34


6.2.2 ResultsResults of the fits to the full flux histories are summarized in Table 6-1 and shown in Figures6-1 through 6-15. As observed in Section 2.1, the permeabilities resulting from the fits fall in areasonably narrow range, typically of order 10-21 mz. The diffusivities, however, are widelydistributed, with most in the range 10-1oto 10-~ m2/s, reflecting variability in the time evolution ofthe seepage from hole to hole. The uncertainties, too, are typically greater for the diffusivities thanfor the permeabilities, again reflecting significant departures of the flux histories from the modelbehavior.The parameter-estimation code ESTIM returns error estimates for the permeability and thediffusivity in the form k + Lk and c ~ kc, respectively, as noted in Section 6.1. The errorestimates for the capacitance, A~, recorded in Table 5 are computed from:(49)Note that the analysis reported here treats the brine density, brine viscosity, and borehole radiusand length as known constants; (i.e., the error estimates do not include the contributions due touncertainty in these parameters). However, such contributions are small compared to theuncertainty in the flux and volume measurements, and the ability of the idealized model to matchthe data.An additional measure of the average properties of the salt, as well as a measure of uncertaintyfor several holes as a group, can be obtained from a fit that lumps the data for a number of holes.This has been done for all the 4-inch-diameter (cl= 0.051 m) holes in Room D, with the exceptionof DBT 14, which exhibited an increasing flux over most of the observation period. Data forDBTIO, DBT11, DBT12, DBT13, DBT15A, DBT15B, DBT3 1A, and DBT32A were included,and eight zero-flux values at early time were removed, in accord with the discussion in theforegoing section. The fit was then performed on the remaining 582 points. The results are:k = (0.57 fO.021) x 10-21m’2 ,c= (9.82* 1.04)x 10–l Omz/s ,C = (0.36 i 0.040) x 10-9Pa-l ,The tit is compared to the data in Figure 6-16,35


Table 6-1. ParametersDerived from Fits to Full Flux.Perm.x Press.Permeability*DiffusivityCapacitance*Holek Pm(m2 Pa, x 10-15)k(m2, x 10-21)(m2/s, ~ 10-10)c(Pa-l x 10-9)DBT 103.84 + 0.240.38 i 0.0240.47 + 0.0783.87 t 0.69DBT1114.92 A 0.551.49 i 0.05535.09 t 6.290.20 * 0.037DBT128.40 * 0.940.84 * 0.094101.73 * 65.330.039 i 0.025DBT 132.29 ~ 0.350.23 + 0.0350.59 i 0.231.85 * 0.77DBT14A10.25 f 3.131.02 i 0.31278. 10* 456.790.018 + 0.031DBT14B29.26 i 3.672.93 * 0.37433.61 t 329.030.032 f 0.025DBT15A4.23 + 0.720.42 i 0.0721.85 t 0.861.09 * 0.54DBT15B2.38 i 0.770.24 i 0.0771.27 t 1.220.89 i 0.90DBT31A11.76 *3.1O1.18+0.314.05 ~ 3.361.38 + 1.20DBT31 c0.037 * 1.340.0037 to. 1340.0034 * 0.2455.24 i 422.59DBT32C0.12 t 3.180.012*0.3180.0045 f 0.23313.07 + 760.22L4B010.88 t 0.560.088 * 0.0560.58 ~ 0.910.73 * 1.24L4X01——QPBO 118.76* 1.181.88+0.118110.04 t 34.070.081 * 0.026QPB02321.34 t 1.2532.13*0.1211.58 i0.1413.22 t0.17QPB0319.06 + 5.931.91 t 0.596388 t 188390.0014 t 0.0042QPB04———QPB05————* Assumes pm = 1.0 x 107 Pa, ~ = 2.1 x 10-3 Pa-s.Holes DBT 16 and DBT 17 yielded no brine, and are not included in the table. Blank entriesindicate failure to obtain a fit.A similar exercise grouped data from all holes in the Room Q Access Drift, with the exceptionof QPB02, which produced brine at a rate one to two orders of magnitude greater than nearbyholes. The fit combines data from QPBO 1, QPB03, QPB04, and QPB05. Data fort> 3x 107s, aswell as 30 zero-flux values at early time, were deleted, leaving 112 points. The results are:k = (0.51 tO.034) x 10-21 m’2 ,c= (6.44* 1.48)x 10–l Omz/s ,C = (0.37*0.090)x 10-9Pa-1 .36


16.014.012.0.-T0;10.0~8.00;IL 6.04.02.00.0kI 1 1 1 I I I I0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Time (s, X107)TRI-6119-281.OFigure 6-1. Comparison of model fit to data for full flux history, hole DBTIO,16.014.012.0=: 10.00&~8.0~LL 6.04.02.00.0’0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Time (s, X107)TRI-6119.282-oFigure 6-2. Comparison o! model fit to data for full flux history, hole DBT I I.37


8.07.06.05.004.0 “3.0 -0002.0 -0001.0- 0uI 1 I I I I 1o.o@0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Time (s, X107)TRI.61 19.283.0Figure 6-3. Comparison of model fit to data for full flux history, hole DBT12.18.016.014.0-_ 12.0; 10.0~~u-8.06.04.02.0)?o0jkdM%z?_oo 00.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0I I I I 1 I I0Time (s, X107)TRI-61 19-284-0Figure 6-4. Comparison of model fit to data for full flux history, hole DBT 13.38


8.07.06.0.--055.00~ 4.0g~ 3.00(J000 o“u0o0 0002.01.00.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0Time (s, X106)TRI-6119.285-OFigure 6-5. Comparison of model fit to data for full flux history, hole DBT 14A.25.020.0: 15.0G~~ 10.0LOm00o@ Ooooo& 000000oOo00 0‘m00 c)00005.000 000.0nn I I I I I I I I0.0 5.0 10.0 15.0 20,0 25.0 30.0 35.0 40.0 45.0 50.0Time (s, X106)TRI-6119.2E6.OFigure 6-6. Comparison of model fit to data for full flux history, hole DBT 14B.39


14.0 -12.0 -z; 10.0 -0G~8.0 -0$L6.0 - 0 0004.0 -02.0 -0.0I I I I0.0 5.0 10.0 15.0 20.0 25.0Time (s, X106)TRI.61 19-287.0Figure 6-7. Comparison of model fit to data for full flux history, hole DBT 15A.20.01 I I 1 I 1 I 1 I18.016.014.0o3z 12.0z~ 10.0~~ 8.0L6.04,02.0000 00I I I 1 10.0“0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.00Time (s, X106)TRI.6119-288-oFigure 6-8. Comparison of model fit to data for full flux history, hole DBT 15B.40


20.018.0 -16.0 -14.0 -& 12,0 – o50-- 10.0 -0go~ 8.0 - 0L6.0 -04.0 -2.0 -1 1 1 4 1 I0.00.0 2.0 4.0 6.0 8.0 10.0 12.0 14,0Time (s, X106)TRI-6119-289-oFigure 6-9. Comparison of model fit to data for full flux history, hole DBT3 1A.20.0 1 I 1 1 0 1 I 318.016.014.0cG-: 12.0x$10.0$ 8.0 -L6.0 -4.000002.0000I ! 1 1 ()0.00.0 4.0 8.0 12.0 16.0 20.0 24.0Time (s, x 1013)TRI.61 19.290.0Figure 6-10. Comparison of model fit to data for full flux histo~, holeDBT31 C,41


8.07.0o6.0.-- 0x5.0~ 4.0g$~ 3.02.0t\L“’o1.0 000 Cg&@5& ‘o%o~o o0.0 II 1 1 10.0 1.0 2.0 3.0 4.0 5.0Time (s, X107)0TRI.6119-291-OFigure 6-11. Comparison of model fit to data for full flux history, hole DBT32C.4.03.00‘7 o;- 2.0~x 3ii1.0000 uot0.00 5 10 15 20Time (s, X106)TRI.61 19-292.0Figure 6-12. Comparison of model fit to data for full flux history, hole L4B01.


30.0 [ 1 I 1 I I 1 127.024.0x~ 15.0goii o~(-JO9.0 -0 06.0 –3.0 t(10.0 II I I I I I I0.0 1,0 2.0 3.0 4.0 5.0 6.0 7.0Time (s, X107)ooTRI-61 19.293.0Figure 6-13. Comparison of model fit to data for full flux history, hole QPBO1.12.011.01 0.09.0 ~~8.0)6.0 -4,0 -3.0 -2.0 -00001.0–000.01 I 1 I f? I 1 00, n I0.0 2.0 4.0 6.0 8.0 10.0 12,0 14.0 16.0 18.0 20.0Time (s, X106)TRI-6119.294-0Figure 6-14. Comparison of model fit to data for full flux history, hole QPB02.43


10.09.008.07.00-~ 6.0x~ 5.0g~ 4.0L3.02.01.00000o0000000 Q)omo%0.0I I 10.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0Time (s, X107)TR1.6119.295-OFigure 6-15. Comparison of model fit to data for full flux history, hole QPB03.20.018.016.014.012.010.06?1>m .#,’I“Q .- #=I II DBTIO @ DBT15BIA DBT1l ■ DBT31 AI o DBT12 ● DBT32A0 DBT13 — Model FitI v DBT15A8.06.04.02.00.00 1 2 3 4 5 6 7 8Time (s, xl 07)TRI-61 19-296.0Figure 6-16.Comparisonof model fit to data for full flux history, Room D holes grouped.44


The fit is compared to the data in Figure 6-17.6.3 Fits Based on the Full Cumulative Volume HistoriesThis section describes fits based on the cumulative volume history for each hole (Equation 36).This exercise offers two advantages over fitting data for the flux. First, the raw data are in theform of the volume collected in each sampling. Thus, data for cumulative volume versus time canbe constructed directly. In contrast, assembly of data for the flow rate involves some sort ofapproximation (see Section 6.2). Second, the data for cumulative volume are much smoother thanthose for the flow rate. The cumulative volume is, by definition, a non-decreasing function, whileapproximations to its derivative with respect to time can be highly variable.40.036.00 QPBO132.028.0A QPB03+ C2PB04X QPB05~ 24.0LG 20.0g~ 16.0L12.00?0–0 o0— Model FitAAA8.0O.OO4.0


Fits to the flux and to the cumulative volume for the same borehole will yield somewhatdifferent parameter values, in part due to the different weights given to different portions of thedata in the two schemes. In particular, a fit to the cumulative volume tends to weight the late-timedata (when the volume is large), while a fit to the flux tends to weight the early-time data (when theflux is large).The fits to the cumulative volume data were again performed with the parameter-estimationcode ESTIM (see Section 6. 1). A subroutine was written to evaluate Equation (36), again leavingtwo parameters to be determined. As is the case for the flux (Section 6.2), there is one parameterthat characterizes the time when the flow evolves, ?0 = a2 /c, and one parameter that characterizesthe magnitude of the volume (per unit area), VO= kpma / PC = Cpma. With a and w known, onemay fit for the diffusivity, c, and the product of the capacitance and the initial pressure, Cpm. Theintegral in Equation (36) was evaluated by Gauss quadrature using the SLATEC subroutineDGAUS8.Hole DBT 15 was elongated after about 295 days. A single fit of the cumulative volume modelto the data for DBT 15 was performed by assuming that one set of parameters characterizes theinitial hole length, and a second set characterizes the additional length. The initial length isassumed to continue producing brine after the extension, and production from the new section issuperposed. Thus, four parameters were determined in a single fit, reflecting the magnitude andthe time scale of the flow to each section of the hole.6.3.1 ResultsFits were obtained for the entire monitored histories of holes DBT 10, DBT11, DBT 12,DBT 13, DBT 15, L4B01, and QPBO 1. For reasons discussed in detail below, fits could not beobtained for the full data available for DBT 14 and QPB02. In these cases, fits were performed tothe data for early time. The remaining holes, DBT31, DBT32, L4X01, QPB03, QPB04, andQPB05, exhibited behavior sufficiently erratic that fits could not bc obtained using the fullhistories.The fits of Equation (36) to data for cumulative volume are shown in Figures 6-18 through 6-26. The resulting parameters are summarized in Table 6-2. The fit determines c and pmC; thecapacitance is determined from the latter, assuming pm = 107 Pa. The permeability k thendetermined using k = WCC, with p. = 2.1 x 10-3 Paos. The diffusivity characterizes the rate ofdecay of the flux; consequently, it is sensitive to the evolution of the flow in time, which is quite46


o.o~I I I 1 I I I J0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Time (s, X108)TRI-6119-298-oFigure 6-18.Comparison of model fit to data for cumulative volume history, hole DBTIO.0.70.60.50.40.30.2 -0.1o.o~1 1 I 1 1 I I0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Time (s, X108)TRI.61 19.299-OFigure 6-19. Comparison of model fit to data for cumulative volume history, hole DBT 11.47


0.300.25000.200.150.100.050.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Time (s, X108)TRI.611 9-300-0Figure 6-20.Comparison of model fit to data for cumulative volume history, hole DBT12,0.20N-0x‘k_0.150.100.05 –1 ! I I I I [o.oo~“;0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Time (s, X108)TRI.61 19-3010Figure 6-21.Comparison of model fit to data for cumulative volume history, hole DBT 13.48


0.100.08m:: 0.060(c)‘&Ez 0.04s0.020.0001 I I 1 I I I t I0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20Time (s, X108)TRI.61 19-302.0Figure6-22. Comparison of model fit to data for cumulative volume history, hole DBT 14A.0.50 1 I I I I I I0.45 -0.40 -$70.35: 0.30 -m-g o,25E2 0.20 -s0.15 -0.100.05 -0.00 (0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Time (s, X108)TRI.611 9.303.0Figure6-23. Comparison of model fit to data for cumulative volume history, hole DBT 15.


0.080.070.06cG-O 0.05G‘E 0.04;% 0.03>0.020.010.000.00 0.05 0.10 0.15 0.20 0.25Time (s, X108)TRI.611 9-304-0Figure 6-24.Comparison of model fit to data for cumulative volume history, holeL4B01.0.16 1 I I I 1 I I I I I I0.14 -0.12 -C$$ox0.10 -0.08 -0.06 -0.04 -0.02 -0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24Time (s, X108)TRI.61 19.305-0Figure 6-25. Comparison of model fit to data for cumulative volume history, hole QPBO 1.50


3.5 I3.0 -Jo c1 (3I I I---0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24Time (s, X108)TRI-61 19-306-0Figure 6-26. Comparison of model fit to data for cumulative volume history, hole QPB02.variable from hole to hole. Correspondingly, the fits yielded a wide range of apparentdiffusivities, of the order of 10-10 to nearly 10-7 mz/s. Similarly, the inferred capacitance rangesfrom 10-11 to IO-8 Pa-1. The permeability is determined from the magnitude of the flow rate,which, as noted above, was observed to fall within a fairly narrow range. Thus, the fits yieldedpermeabilities typically in the range 1O-ZZto 10-21 mz, with the exception of QPB02, which is ofthe order of 10-20 mz.6.4 Fits Based on the Late-Time Flux HistoriesAn attempt was also made to fit the late-time, asymptotic solution for the flux (Equation 43) tothe data. The disadvantage of this approach is that it discounts the early-time response, when theflow evolves relatively rapidly, and the data contains information more sensitive to the diffusivity.This limitation is manifested in the difficulty of obtaining a reliable value for the intercept B, asdiscussed in Section 5.4. However, there are advantages to considering the late-time response, inaddition to its simplicity. First, plots of lql–’ versus In t give some visual indication of how far the


Table 6-2. Parameters Derived from Fits to CumulativeVolumeDiffusivity Perm. x Capac. Capacitance* Permeability*Hole c pmc c k(m2/s, x 10-10) (—, x 10-2) (Pa-l, x 10-9) m2, (x 10-21)DBTIO 3.11 1.03 1.03 0.672DBT11 57.7 0.133 0.133 1.61DBT 12 141 0.0288 0.0288 0.853DBT 13 1.50 0.956 0.956 0.302DBT14At 133 0.0293 0.0293 0.819DBT 14B — —DBT15A 4.28 0.680 0.680 0.612DBT15B 0.660 1.33 1.33 0.184DBT3 I — —DBT32L4B01 44.0 0.185 0.185 0.171L4X01 — — —QPBO1 536 0.0212 0.0212 2.39QPB02* 9.62 14.4 14.4 29.1QPB03QPB04QPB05 —* Assumes pm = 1.0 x 107 Pa, p =2.1 x 10-3 Pa-s.t Fit fort< 8.64 x 106 s (100 d).~Fit fort< 1.04 x 10Ts(120 d).idealized model can be taken, and where it begins to break down. A linear portion of the curvewith positive slope is the region where the model may yield meaningful results. A departure fromthe linear trend at some point may indicate that phenomena not accounted for by the model, such ascrack growth, started to influence the flow. Second, this method allows a local fit in those regionswhere it appears to apply; for example, data at very late time that become erratic can be excludedfrom the fit. Of course, this introduces considerable subjectivity into the fitting process, and theresults should be viewed with this in mind.52


Note that the data for hole DBT 15 were fit in this case as two independent sets. The data forthe first stage, DBT1 5A, were fit for properties representative of the salt penetrated by the initiallength of the hole. For the purpose of fitting the second stage following elongation, DBT 15B, allbrine collected was assumed to be derived from the new length of hole. That is, for the late-timeanalysis, the contribution of the initial hole length to the brine collected during the second stagewas considered negligible.6.4.1 ResultsFigures 6-27 through 6-40 show all data plotted as lql-’ versus in tand the linear fits found forselected portions of the data. The resulting hydraulic properties are given in Table 6-3. Again, thepermeability values are more reliable than the capacitance values due to limitations discussedpreviously. This fact is borne out by the relatively narrow range of permeability inferred, and thereasonable agreement with results from the more complete fits given in Tables 6-1 and 6-2. Thepermeabilities inferred here are again of the order of 10-22 to 10-21 mz, with one (QPB02) of order10-’20 mz. Most are within a factor of 2 of the values obtained by fitting the entire flux orcumulative volume histories.The diffusivities or capacitances inferred by this method must be regarded as questionable.The latter fall in a range of order 10-Iz to 10-10 Pa-1, with one (QPB02) value of order 10-9 Pa-1,and DBT12 giving C = 8.5 x 10-16 Pa-l. The latter value is clearly absurd, as it impliescompressibilities many orders of magnitude less than can be rationalized for the salt and brine (seeSection 7.6). The extremely small capacitance inferred in this case results from the very slowdecay rate observed at late time.Notably, the capacitances inferred from these fits are, in many cases, about an order ofmagnitude smaller than those from the full fits. This occurs because the full data for many holesexhibit fluxes that drop off very slowly at very late time. Thus, the fits to the full data sets arrive ata relatively large value of C in order that the characteristic time for the decay of the flux be large.In contrast, because the fits of the asymptotic expression (Equation 43) typically exclude the verylate time data, when the linearity of lql-] versus in t appears to break down, intermediate times(typically centered around - 10Ts) are given more weight. During the period when the asymptoticexpression appears to represent the data, the decay of the flux occurs at a somewhat higher ratethan that inferred for the full histories using Equations (35) or (36), and this is reflected in smallerinferred values of C.53


6.05.0(_)0- 4.0-0xE 3.02xc?2 2.01.00.0, ~6,07 108Time(s)TRI.6119.307-0Figure 6-27.Comparison of model fit to datalinear regression are shaded.for late-time flux, hole DBT 10. Points included in40.0 , ,,, ,,, ,, 1 ,,! 1 1 1 1 I,!, 1 1 I35.0 - 0()30.0 - ~oc.)o.o~ ‘ “ ““1 1 1 11I 1 !11 I1[,05 106 ,07 108Time(s)TR161193080Figure 6-28.Comparison of model fit to datalinear regression are shaded.for late-time flux, hole DBT 11. Points included in54


10.0 ,,, ,, 1 I 1 I ! 1 I 1 1 1 1 1 I I 19.0 -8.0 ‘-00-7.0 - 0‘Z 6.0 -xE- 5.0 -m; 4.0 – o.-------O-J.*.%3.0 -. .. . ..-------- .. . . .. .-2.0 -1,0 -o.0 ~g~.;clo~[) o&00 o-x)~ () o G0.0, I I I 1 1 1 1 I 1 1 I ( 1 1 1, ~6 ,07 ,08Time(s)TRI.61 19.309.0Figure 6-29. Comparison of model fit to data for late-time flux, hole DBT12. Points included inlinear regression are shaded.10.09.08.07.00-‘~ 6.0xk- 5.0In$ 4.03.02.01.00.01 06 ,07 108Time(s)TRI.6119.31 O.(JFigure 6-30. Comparison of model fit to data for late-time flux, hole DBT13. Points included inlinear regression are shaded.55


x2ii50.0 1 1 1 I 1 111 I I j, I 1 1 1II-.,..45.0 I,. .. ..., 040.0 - .,0 .“35.0 -30.0 - ●* @o25.0 -20.0 - ●’”’-.,”.,15.0 Q.’”’.,,10.0 ~“”..”0,.”‘“”;● ..=” ~,S”” o 0 0o..’% (9° ~ “~ooO@ o 08 0,.-5.0 tTime (s)0.0! ! I 1 111I 1 1 1 I I I,I,06 ,07 ,08TRI.6119-311OFigure 6-31. Comparison of model fit to data for late-time flux, hole DBT 14A. Points included inlinear regressionare shaded.40.0 1 ,1 1 1 II, I 1 ,.”1 1 1 1,..35.0 -30.0●,.’””..“”... .“0;x25.0 -E> 20.0 - u~“?“P;●4”’00x 9“ii..415.00 (1●“*,.”.,,”10.0 ~~ o ~).,“..’,.”5.0 - ,.”,.,.’,,,I I I I I I , , I0.0106 ,07 ,083 00Time(s)TRI-61193120Figure 6-32. Comparison of model fit to data for late-time flux, hole DBT 15A. Points included inlinear regressionare shaded.56


8.07.06.00-‘05.0E-34.03.02.01.00.0II I I 1 1 I I I I I 1 , , I 1 I 1,06 ,07 108Time(s)TRI.6119-313-OFigure 6-33.Comparisonof model fit to data for late-time flux, hole DBT 15B. All points shownwere included in linear regression.20.018.016.0m;14.0~ 12.0E“~ 10.0xi? 8.06.04.02.00.0106 ,07 108Time(s)rR1.ljl 19-314-0Figure 6-34.Comparisonof model fit to data for late-time flux, hole DBT31 A. All points shownwere included in linear regression.57


0--0xE2x3ii20.018.016.04.02.00.08.06.04.02.0.’.’,,”.d-0 .-,.”,.,..,●..”” .,.4°””● 0 /’ 0°.“e.,”4 0‘m..’0,. 0 0° m““o..”,.-,,>. 0,.-,,,,. c),,.“’ ,. 0c)0.0 .10’Time(s)TRI-6119315.OFigure 6-35.Comparison of model fit to data for late-time flux, hole DBT32C. Points included inlinear regressionare shaded.12.010.00- 8.0-0ẋ~- 6.0mxe~ 4.02.00o00,.,.“’. ,,,.”. .“,.’,.,y”.,m”# .ooo ““’P.““’”’ ●●.”.“,.● .’” ,..,“y’”0.0106 ,07Time(s)TRI.6119.316.OFigure 6-36.Comparison of model fit to data for late-time flux, hole L4B01. Points included inlinear regressionare shaded.


40.0 1 1 I 1 r I 1 1 I35.0 -030.0 -0-‘0 25.0xE 20.02x< 15.0:110.0.“%,. s“”●,.,.0”●0,.””;... ●5.0 - .,.,,..,.-,.,1 I I I 1 ! 1 I0.0,06 ,07.“,.o,...✎✌● ✎✎✎✎✌✌✎✌✌✎✌..’Time(s)TR16119-317.OFigure 6-37.Comparison of model fit to data for late-time flux, hole L4X01. Points included inlinear regression are shaded.16.0 1 I ! I I I 1 I I 1 I I I 1 1 1 1 1 , rrgx14.0 -12.0 - ● ...’...●,...10.0 -E so _>’x3ii 6.0 -Lloc1“%.:.8:m.0%“”’ ~,.,/....’O...4.0 - 0 0 0 =“” o2.0 - ...,,,.....-... .,.,-..w’.,’...,.-..II0.0 ““”.,05 106 ,07 ,08Time(s)TR1611931E0Figure 6-38.Comparison of model fit to data for late-time flux, hole QPBO 1. Points included inlinear regressionare shaded.59


20.0 I I 1 1111 II 1 1 1 ,, 1 1 II 1 1 1 1 I 1 1 1 I 1 ,! 118.0 -16,0 -14.0 -&i;~ 12.0 -0E~ 10.0 -x3K 8.0 -6.0 - -----(_) -....”-”””...-”4.0 - A,@*-**2.0 –~ .-.9-..0UI I 1 1 1 1 11 .. .“ 1 ! ! 1 1 1 I I I 1 1 I I 1 I I 1 1 I 1 , 1 lb0.0,04 ,05106 ,07 108Time(s)TR16119319-0Figure 6-39.Comparison of model fit to datalinear regression are shaded.for late-time flux, hole QPB02. Points included in25.01 I 1 1 [ 1 1 1I 1 I 1 I111[ I 1 I 1 1!11I20.015.010.0● 0..- ..-..--”● ....””... ....● w“””” ●.9-” 0.●%“”● .O..”””””- # ●0 ‘o... ●..-..--”..- ●..- ... ..-” ● .●..- 0..--”05.0t I I 1 1 !,1I 1 1 1I!,II I I 1 I 1 , I0.0105 106 ,07 108Time(s)TR161 193200Figure 6-40,Comparison of model fit to data for late-time flux, hole QPB03 Pointsincludedinlinear regressionare shaded.60


Table 6-3. Parameters Derived from Fits to Late-Time FluxPerm. x Press. Permeability* Diffusivity Capacitance*Hole k Pm k c(m2 Pa, x 10-15) (m2, x 10-21) (m2/s, ; 10-10) (Pa-l, x 10-9)DBT 10 8.33 0.833 22.8 0.174DBT11 20.7 20.7 508 0.0195DBT12 26.0 26.0 1.45 x 107 8.53 X 107DBT 13 4.38 4.38 22.8 0.0914DBT 14A 4.72 4.72 49.4 0.0455DBT14B —DBT 15A 4.58 0.458 11.5 189DBT15B 12.8 1.28 133 0.0457DBT31 A 12.2 1.22 29.8 0.195DBT32C 3.31 0.331 154 0.0102L4B01 1.40 0.140 13.6 0.0492L4X01 3.44 0.344 626 0.00261QPBO 1 12.6 1.26 42.6 0.141QPB02 287 28.7 41.3 3.31QPB03 15.6 1.56 4300 0.00172QPB04 — —QPB05 —* Assumes pm = 1.0 x 101 Pa, ~ =2.1 X 10-3 Pa-s.61


7.0 SUMMARY AND DISCUSSIONData from the small-scale brine inflow experiments for the period to January 1990 have beenreduced and analyzed. The permeability and hydraulic diffusivity of the salt have been estimatedby fitting an idealized model to the data. The model assumes that brine seepage to the monitoredboreholes is due to relaxation of ambient pore-water pressure by darcy flow. It is further assumedthat the salt is isotropic and homogeneous around each hole, the initial pressure is uniform, and theflow is normal to the hole axes. An exact, analytical solution is available for the linear diffusionproblem corresponding to this configuration. The solution has been fit to the data for each hole bya least-squares method, which yields estimates of two parameters for each fit. The two parametersscale the magnitude of the flow and the time scale over which it evolves. With an additionalassumption for the initial brine pressure, the permeability and diffusivity can be extracted.Three fitting schemes have been applied, entailing fits to the brine flux, the cumulative brinevolume, and the brine flux at “late” time. For those data consistent with the model prediction,estimated permeabilities are typically 10-22 to 10-21 m2. The relatively small range ofpermeabilities inferred reflects the observation that the observed seepage fluxes are fairly consistentfrom hole to hole, of the order of 10-10 m/s. Estimated diffusivities are typically 10-10 to 10-8m2/s. The great scatter in inferred hydraulic diffusivities is due to the difficulty of matching theidealized model history to the observed evolution of the flows. The data obtained from several ofthe monitored holes are not consistent with the simple model adopted here; material propertiescould not be inferred in these cases.The results of the fits to the full flux histories are displayed graphically in Figures 7-1 through7-3 as histograms of the permeability, hydraulic diffusivity, and capacitance, respectively. A totalof 15 fits are available (see Table 6-1). The histograms are constructed with logarithmic scales forthe properties. The “bins” used to construct the histograms are bounded by (0.5 - 5.0) x 10n,where 10n is plotted at the center of each bar. The logarithmic mean of each of the three propertiesis:~=0.48x 10-*1 m2 ,F= 5.34x 10–10 m2/s ,C= O.43X1O-9 Pa-1 .The range spanning plus or minus one standard deviation of the logarithm of each parameter is:~=(0.056– 4.1)x 10-21 m’2 .z = (0.075 – 380) x 10–10 m2/s ,~ = (0.029 – 6.4) x 10-9 Pa-l .63


8765Z43211 ()-24 10-23 10-22 , ()-21 , ()-20 ,0-19k (m2)TRI.6119.321-OFigure 7-1. Histogram of permeabilities; fits to full flux histories (Table 6-1). Bins centered on10n include values from 0.5 to 5.0 x 10n. Mean value of log k is -2 1.3; standarddeviation of log k is 0.9.64


8761524.,.-12-11 ,..10 ,0-9 ,0-8 ,0-710C (Pa-l)TRI.61 19.323.0Figure 7-3. Histogram of capacitances; fits to full flux histories (Table 6-1). Bins centered on10n include values from 0.5 to 5.0 x 10n. Mean value of log C is -9.4; standarddeviation of log C is 1.2.The present data are too sparse to draw conclusions regarding the statistical distributions of theproperties. Notably, it is sometimes claimed that permeability is log-normally distributed within agiven geological formation (Freeze and Cherry, 1979). Figure 7-1 does not contradict this claim.Freeze ( 1975) found that the standard deviation of the logarithm of permeability typically falls inthe range 0.5 to 1.5. The standard deviation of the logarithms of the 15 permeabilitydeterminations shown in Figure 7-1 is 0.9, in remarkable agreement with the observation of Freeze(1975), based on much more extensive data sets. Thus, the scatter in the permeabilities inferred inthis study appears to be typical of geologic media.It should be noted that the boreholes in the Room Q Access Drift, QPBO 1 through 05, allpenetrate Marker Bed 139, a horizon known to produce relatively large quantities of brine (Finleyet al., 1992). The average thickness of the marker bed is about 0.9 m; the QPB holes are 3.1 mlong. Therefore, the brine inflow to these holes may be interpreted as being dominated by themarker bed, and the fits as characterizing the properties of the marker bed. In this case, accordingto the discussion in Section 5.5, both the permeabilities and the capacitances given in Tables 6-1,6-2 and 6-3 should be multiplied by a factor of 3.4.65


Some of the limitations of the idealized, one-dimensional, unbounded, radial flow model usedhere are discussed in the reports by Webb (1992) and Gelbard ( 1992).7.1 Choice of Parameters for the FitsThe regressions reported here are two-parameter fits. For example, as noted in Section 6.1,the flux is fit for a magnitude, qO= kpm / pa, and a characteristic decay time, to = a2 /c. Theapproach followed in the foregoing was to treat the borehole radius, a, and the brine viscosity, ~,as known, because they can be determined with good accuracy by direct, independentmeasurements. The initial brine pressure was also treated as known, with the constant value pm =10 MPa assumed. This leaves the permeability, k, and the diffusivity, c, as unknownsdetermined by the fits. The capacitance, C = k/ vc , can then be computed as well.An alternative scheme that can be applied in a system whose mechanical properties are wellcharacterized is to consider the capacitance known. This requires a model for the relationshipbetween capacitance and various material properties (e.g., Equation 15), as well as independentlaboratory measurements of the rock and fluid compressibilities. In this case, the two remainingunknowns determined by the fits are the permeability and the initial pressure, pm. This scheme isattractive because pm is indeed unknown, can be expected to exhibit some variability, and can bedetermined only by in situ measurements. However, its success rests on the assumption thatthestorage mechanisms are well understood, properly represented by the model adopted, andcharacterized accurately by the independent measurements. That this is not the case for the salt issuggested by the following argument. The field data for borehole seepage considered in this reportindicate a relatively long time scale that characterizes the decay of the flux, typically 100 days. Thelong time scale implies that the diffusivity is small, which, for fixed capacitance, implies that thepermeability is small. In order to match the magnitude of the flux given a small permeability, theinferrred formation pressure must be unrealistically large.A sample calculation illustrates this point. Suppose that to is 107 s (-116 days); assume thatC = 1.0x 10-11 Pa-], which is consistent with various estimates based on the elastic properties ofthe salt and brine (Freeze and Cherry, 1979; Nowak and McTique, 1987). Then, with a = 0.05 mand ~ = 0.0021 Pas, the inferred permeability is k - a2p.C/to =5 x 10-24 mz. That gives (withgo ~ lo- ‘0 ds), pm - qop.a/ k G 2 x 109 Pa, which is clearly absurd (the lithostatic stress is two66


orders of magnitude smaller). Thus, as discussed in the foregoing paragraph, this scheme leads toimplausible estimates of both the permeability and the formation pressure.The fitting scheme adopted in this report, (i.e., to regard the formation pressure as known andthe permeability and capacitance as unknowns), in some sense acknowledges that the mechanismsaffecting capacity or storage in the salt may not be understood fully. This uncertainty is thensubmerged in the estimation of the capacitance, which should be viewed as an “effective” or“apparent” property of the salt. It is important to recognize, however, that the large values ofcapacitance inferred from the fits (e.g., Table 6-1) are not easily reconciled with the classicalmodels of an elastic, porous skeleton and compressible fluid (e.g., Equation 15). Such modelspredict a capacitance of the order of C -10-11 Pa-1, while the fits yielded many estimates of theorder of 10-10 to 10-8 Pa-1. These estimates are discussed further in Section 7.5.7.2 Validity of the Radial Flow ApproximationAfter a sufficiently long time, the measured flow to a borehole becomes sensitive to the finitelength of the hole. A simple validity check for the one-dimensional, radial flow assumption is tocompute the “diffusion” length for the duration of the test, and compare this to the length of thehole. The diffusion length is defined by Ld = V, and represents the characteristic radial distanceto which the pressure relaxation has propagated after time t. If Ld / L


Table 7-1. Diffusion Length Scales for Various Diffusivities1I—Diffusion Length ( Ld = dct, m)Diffusivity ( c, mz/s) t=2.5x107s r=7.5xlo7s10-10 0.050 0.0871()-90.158 0.2741()-8 0.500 0.86610-7 1.58 2.7410-6 5.00 8.66where the non-dimensionalizations are the same as those that were introduced in the cylindricalcase (see Section 5). Thus, the effect of the third dimension is to allow the flux to decay morerapidly, and to approach a non-zero, steady-state value. If a particular data set is influenced bysuch geometric effects, but is fit by the radial flow model, the fit will tend toward small values ofthe diffusivity in order that the flux “hold up” over a long period of time.Additional multidimensional effects are introduced by the presence of the room from which aborehole is drilled. First, there is seepage toward the mined faces of the room. Second, the initialmean stress state penetrated by a borehole is influenced by the room; the mean stress is reducednear mined faces,and the initial, undrained brinepressureisconsequent y reduced as well.Recentwork by Gelbard (1992) considers both seepage toward the room wall and a depth-varying initialbrine pressure in the context of the model discussed in this report. Again, both effects tend toweaken the brine inflow to a borehole. Neglect of these effects, as in the analysis presented in thisreport, results in underestimates of the permeability. Finally, creep of the salt results in acontinued evolution of the mean stress field in the neighborhood of mined cavities, and this too iscoupled to the brine flow. This effect is also neglected in the idealized analysis presented in thisreport.7.3 Comparison of Fits Over Various Time IntervalsThis section provides a comparison of parameters derived from fits of both the flux andcumulative volume models to the same data over various time intervals. This exercise gives thereader a measure of the sensitivity of the results to various possible fitting schemes. Only data68


from DBT 10 are treated; this is one of the “better-behaved” data sets, insofar as the response of thehole was quite smooth, and the rate of decay of the flux decreased monotonically. Results aresummarized in Table 7-2. The variation in the inferred material properties reflects comments madepreviously in this report. The permeability is reasonably well constrained; all values Fall withinabout a factor of 2 of k G 10-21 m2, whether from a fit to the flux or the volume, and also whetherthe data span 830, 470, or 230 days. The inferred diffusivity, however, is one to two orders ofmagnitude larger for the cumulative-volume fits, and it increases significantly for fits to the shorterhistory. Recall that the fits to the flux are more sensitive to the early-time response, so that theinclusion of more data has relatively little influence, here decreasing the inferred diffusivity by onlya factor of 2. In contrast, the fits to the volume are more sensitive to the later response, and thecontinually decreasing apparent diffusivity with longer data sets reflects the fact that the observedflow decayed more slowly than predicted by the idealized model at late time. That is, thecharacteristic time for the decay of the flux appears to be increasing with time. Therefore, a fit to alonger data set arrives at a smaller value of the diffusivity, c.Table 7-2. Parametersfor DBT 10 by Various Fitting SchemesBasis of fitDiffusivityData type Time period (d) c(m2/s, x 10-10)flux 8300.54flux 4701.1flux 2301.1cum. vol. 8303.1cum. vol. 47017cum. vol. 23082* Assumes pm = 1.0 x 107 Pa, ~ = 2.1 x 10-3 Paos,Permeability*k(m2, x 10-21)0.410.520.540.671.11.7Capacitance*C(Pa- I, x 10-9)3.62.32.31.00.310.097These results emphasize that the permeability is fairly well constrained simply by the magnitudeof the flows observed, while the diffusivity is more difficult to bound. Determination of thediffusivity depends on the details of the flow history, which was erratic for some holes.Furthermore, the flow histories are more likely to exhibit effects not accounted for by the idealizedmodel for uniform, radial flow.69


7.4 Comparison of Fits by Various MethodsFigures 7-4 and 7-5 provide a graphical comparison of the variation in the parameters inferredby the three fitting procedures described in Section 6 and summarized in Tables 6-1,6-2 and 6-3.In each case, the parameters inferred from fits based on the first method named in the legend areplotted as “Fit 1,“ and those from the second method are plotted as “Fit 2.” Identical results fromeach method, of course, plot on a line with unit slope. Departures from this line indicate thediscrepancies between methods. As noted previously, the fits for permeability generally result inreasonably consistent results, and no systematic variations are apparent in Figure 7-4. The inferredcapacitances shown in Figure 7-5 exhibit larger differences, depending upon the fitting method.The plot shows that the fits based on the late-time-flux approximation tend to result in muchsmaller estimates of the capacitance. The reason is discussed in Section 6.4.102k.’1 I 1 I I Ill I ! 1 1 I 1 11/ I 1 1 1 ) 1 1 II 1 I 1 1 I 10 Full Flux – Cumulative VolumeA Full Flux – Late Flux1010 cumulative Volume – Late Flux— Perfect Correlations0N.—u100OAAAoo,0-1!2,()-2 v , I 1 I 1 ! ! ,1 , I 1 I ! 1 II ! , 1 1 I 1 I II 1 1 1 1 1 1 1 Ll10-2 ,0-1100 101 102Fit 1TRI.6119-324.OFigure 7-4. Comparison of permeability values (mz, x 10-21) determined by various methods.Perfect correlationslie on the solid line.70


D Full Flux – Cumulative Volume10’A Full Flux – Late Flux0 Cumulatwe Volume – Late Flux— Perfect Correlations1O’J❑Cu.—L10-10 A10-210-3 1()-2 10.1 100 10’ 102Fit 1TRI.6119-325.OFigure 7-5. Comparison ofcapacitance values (Pa-l, xl O-g) determined by various methods.Perfect correlationslie on the solid line.7.5 Additional Sources of Uncertainty7.5.1 initial Brine PressureThe formation pressure was assumed to be 10 MPa in order to separate the permeability. Theerror in the estimates of the permeability is proportional to the error in pm. Various independentestimates of pm typically fall in the range of 0.5 to 12 MPa (Beauheim et al., 1991). In general, itappears that pressures of about 10 MPa are found for measurements taken in the “far field” (i.e., atdistances of the order of 10 m or greater away from facility excavations). Measurements in saltwithin a few meters of mined faces typically yield smaller estimates of the pore pressure.Beauheim et al. (1991) inferred formation pressures from pressure-pulse tests in boreholes drilledfrom Room C2 in a setting similar to that of Room D. Pressures in three sections within the firstfew meters below the floor were inferred to be 0.5,3.2, and 4.1 MPa, suggesting that stress reliefdue to room presence causes a significant reduction in pore pressure.


The parameter estimates reported here for the small-scale brine inflow experiments yield valuesfor the product of the permeability and the initial pressure. The permeability has been separatedonly by making the additional assumption that pm = 10 MPa. This choice may be appropriate forsalt undisturbed by excavation effects, but is likely too large for the salt penetrated by the boreholesunder consideration, as suggested above. If a better estimate of pm is available, the estimates of kand C reported here can be resealed accordingly. As noted previously, the adjustment isproportional to the change in the assumed initial pressure. If one takes, for example, pm = 5MPa,the estimated permeabilities must be multiplied by a factor of 2, and the estimated capacitancesmust also be multiplied by 2.7.5.2 Brine DensityThe data reduction reported here assumed a mass density for the brine of 1.2 x 103 kg/m3.Recent direct measurements on samples taken from the study holes indicated 1.224 to 1.240 x 10qkg/m3 (Howarth et al., 1991) Thus, the volume, flux, and permeability estimates are too large by2 to 3~0. Diffusivity is unaffected, but capacitance, calculated from C = k/ wc, is proportionatelytoo large.7.5.3 Brine ViscosityThe viscosity used in the results reported here is 2.1 x 10-3 Pa-s, based on directmeasurements on brines collected in this study. The viscosity measurements were performed witha capillary viscometer, using brines from hole QPB02, over a range of temperatures. The valuecited here is for 28”C. There is, of course, some uncertainty associated with this value. If theviscosity is larger, the inferred permeabilities must be correspondingly larger in order to obtain thesame flow rate at the same pressure gradient. The diffusivity is unaffected, but the capacitancemust be made larger along with the permeability in order to keep the diffusivity the same.Similarly, if the viscosity is overestimated, the permeability and capacitance are proportionatelyoverestimated. For example, if the viscosity is estimated to be known to within Ml. 1 x 10-3 Pa-s,this introduces a corresponding uncertainty in the reported permeabilities and capacitances of about5%.72


7.6 Expected Capacitance for an Elastic Matrix and FluidIt was noted in Section 7.1 that the estimates of the capacitance of the salt inferred in this studyare quite large in comparison to values expected on the basis of classical models for elasticconstituents. Eleven of the fifteen estimates of C shown in Table 6-1 and Figure 7-3 fall in therange 10-10 to 10-8 Pa-1.Estimates based on the elastic moduli of the salt and brine are of the order of 10-11 Pa-1(Nowak et al., 1988; Beauheim et al., 199 1). Parameters that enter into such estimates arerecorded here in Table 7-3 for reference. Use of the moduli shown in Table 7-3 and @o= 0.01 inEquation (15) yields C - 0.8 x 1O-I1 Pa-1. The calculation is rather insensitive to the estimate ofthe connected porosity, $0; for example, the assumption of @o = 0.001 and the same moduliresults in C -0.6 x 10-11 Pa-1.Table 7-3. Elastic Properties of Salt and BrineParameterValueUnitsReferenceDrained Bulk Modulus,K20.7GPaKrieg (1984)Poisson Ratio, u0.25—Krieg ( 1984)Solid Bulk Modulus, K.23.5GPaSurnino andFluid Bulk Modulus,Kf4.0GPaAnderson (1984)*Porosity, +00.004-0.01—Stein and Kimball( 1992)* determined from acoustic measurementsBeauheim et al. (1991) report a more detailed compilation of ranges of properties by lithology.They compute weighted averages of the storage based on the fractions of the specific lithologiespresent in a given test interval. The resulting capacitances fall in the range (0.8 - 1.3) x 10-11Pa- 1, consistent with the calculation based on the values given in Table 7-3.As noted in Section 7.1, the capacitances inferred in this study should be regarded as“effective” or “apparent” values. These fits yield large estimates of the capacitance that are difficultto reconcile with the classical model of storage due to elastic compressibility of the porous skeletonand the fluid. Recall that the large estimates of capacitance are a consequence of the relatively large73


time scale that characterizes the observed decay of the flux. This long time scale possibly arisesfrom some mechanism(s) other than diffusive relaxation of pore pressure. Deformation of the saltin the neighborhood of excavations is one obvious candidate source for other time scales that mayinfluence the brine flow (McTigue et al., 1989). It is known, for example, that the salt is subject torate-dependent creep, crack growth, and other processes that may have a profound influence onbrine seepage. These processes can lead to time-dependent relaxation of the mean total stress CJ,inelastic dilatation of the salt, large permeability changes, imbibition of air and consequentmultiphase flow, etc. Such coupling may introduce time scales longer than the expected diffusivetime scale, which may be reflected in the data.In this context, it is also important to note that the data collected from January 1990 throughJune 1991, which were not treated in this report but are summarized in Finley et al. (1992),indicate fluxes that either level off or increase slowly for most of the holes. This response isclearly not in accord with the idealized model used in this study to interpret the earlier data. Theexplanation may lie with the continued evolution of the state of stress and the microstructure of thesalt surrounding the excavations.74


8.0 ReferencesBear, J. 1972. Dynamics of Fluids in Porous Media. New York, NY: American ElsevierPublishingCompany.Beauheim, R. L., G.J. Saulnier, Jr., and J.D. Avis. 1991. Interpretation of Brine-Permeability Tests of the Salado Formation at the Waste Isolation Pilot Plant Site: FirstInterim Report. SAND90-0083. Albuquerque, NM: Sandia National Laboratories.Blackwell, J. H. 1954. “A Transient-Flow Method for Determination of Thermal Constants ofInsulating Materials in Bulk,” Journal of Applied Physics. Vol. 25, no. 2, 137-144.Crank, J. 1975. The Mathematics of Diflusion. 2nd ed. Oxford: Clarendon Press,.Finley, S. J., D.J. Hanson, and R. Parsons. 1992. Small-Scale Brine Inflow Experiments—Data Report Through 6/6/91. SAND91-1956. Albuquerque, NM: Sandia NationalLaboratories.Freeze, R. A. 1975. “A Stochastic-Conceptual Analysis of One-Dimensional Groundwater Flowin Non Uniform Homogeneous Media, ” Water Resources Research. Vol. 11, no. 5, 725-741.Freeze, R. A., and J.A. Cherry. 1979. Groundwater. Englewood Cliffs, NJ: Prentice-Hall.Gelbard, F. 1992. Exact Analysis of a Two-Dimensional Model for Brine Flow to a Boreholein a Disturbed Rock Zone. SAND92- 1303. Albuquerque, NM: Sandia NationalLaboratories.Hills, R.G. 1987. ESTIM: A Parameter Estimation Computer Program. SAND87-7063.Albuquerque, NM: Sandia National Laboratories.Howarth, S. M., E.W. Peterson, P.L. Lagus, K. Lie, S.J. Finley, and E.J. Nowak,. 1991.“Interpretation of In-Situ Pressure and Flow Measurements of the Salado Formation at theWaste Isolation Pilot Plant,” Proceedings of the Society of Petroleum Engineers RockyMountain Regional Meeting and Low-Permeability Reservoir Symposium, Denver, CO,April 15-17, ]99]. SPE-2 1840. Richardson, TX: Society of Petroleum Engineers, 355-369.75


Jaeger, J.C., 1958. “The Measurement of Thermal Conductivity and Diffusivity with CylindricalProbes,” Transactions, American Geophysical Union. Vol. 39, no. 4, 708-710.me–XU2 du ,,Jaeger, J. C., and M. Clarke. 1942-1943. “A Short Table of ~() J;(u) +Y~(u) 1’Proceedings of the Royal Society of Edinburgh. Vol. A61, pt. III, 229-230.Krieg, R.D. 1984. Reference Stratigraphy and Rock Properties for the Waste Isolation PilotPlant (WIPP) Project. SAND83- 1908. Albuquerque, NM: Sandia National Laboratories.McTigue, D.F. 1986. “Thermoplastic Response of Fluid-Saturated Porous Rock,” JGR. Journalof Geophysical Research B. Vol. 91, no. 9, 9533-9542.McTigue, D. F., S.J. Finley, and E.J. Nowak. 1989. “Brine Transport in Polycrystalline Salt:Field Measurements and Model Considerations,” EOS Transactions, American GeophysicalUnion. Vol. 70, no. 43, 1111.Nowak, E. J., and D.F. McTigue. 1987. Interim Results of Brine Transport Studies in theWaste Isolation Pilot Plant (WIPP). SAND87-0880. Albuquerque, NM: Sandia NationalLaboratories.Nowak, E. J., D.F. McTigue, and R. Beraun. 1988. Brine Injlo w to WIPP Disposal Rooms:Data, Modeling, and Assessment. SAND88-01 12. Albuquerque, NM: Sandia NationalLaboratories.Rice, J.R., and M.P. Cleary. 1976. “Some Basic Stress-Diffusion Solutions for Fluid-SaturationElastic Porous Media with Compressible Constituents,” Reviews of Geophysics and SpacePhysics. Vol. 14, no. 2, 227-241.Rutherford, B. 1992. Appendix F: “Statistical Analysis of Brine Inflow Data,” Small-ScaleBrine Inflow Experiments—Data Report Through 6/6/91. S.J. Finley, D.J. Hanson, andR. Parsons. SAND91- 1956. Albuquerque, NM; Sandia National Laboratories. F-2 throughF-3 1.76


Stein, C. L., and K.M. Kimball. 1992. Appendix D: “Water Content and Insoluble ResidueAnalyses of WIPP Samples,” Small-Scale Brine Inflow Experiments—Data ReportThrough 6/6/91. S.J. Finley, D.J. Hanson, and R. Parsons. SAND91 -1956. Albuquerque,NM; Sandia National Laboratories. D-1 through D-8.Sumino, Y., and O.L. Anderson. 1984. “Elastic Constants of Minerals,” Handbook of PhysicalProperties of Rocks. Ed. R.S. Carmichael, Boca Raton, FL: CRC Press, Inc. Vol. III, 39-138.Webb, S. W., 1992. Brine Inf70w Sensitivity Study for Waste Isolation Pilot PlantBoreholes: Results of One-Dimensional Simulations. SAND9 1-2296. Albuquerque, NM:Sandia National Laboratories.77


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