Methods of interpretation of borehole falling-headtests performed in compacted clay liners
Paul Chiasson
Abstract: The interpretation of falling-head tests in cased boreholes is discussed. These tests are commonly used tomeasure hydraulic conductivity of compacted clay liners and are often part of the construction quality assurance pro-gram. Three methods of interpretation are reviewed with data sets collected from real tests. Two of these methods havebeen the subject of past research by other authors: the Hvorslev, or time-lag, method and the velocity method. Afterthe limitations of these two approaches have been underlined, a third method is proposed. It uses a best linear unbiasedestimator to fit the theoretical head difference function in a plot of falling water column elevation as a function of time(Z–t method). The Hvorslev method is found unreliable and is not recommended. The velocity method is theoreticallysound, but statistical uncertainty can become high when this method is used in testing materials with low hydraulicconductivity, such as clay liners. Materials with low hydraulic conductivity tend to produce scattered velocity plots,creating considerable uncertainty for the estimated k value. The proposed Z–t method is less sensitive to inaccuracies inmeasurements, yielding a more reproducible result. An interpretation method for stages I and II of two-stage boreholetests is also proposed. This method yields the anisotropy of the liner and the vertical hydraulic conductivity. As a resultof inaccuracies in measurements and limited difference between the geometries of stages I and II, the computed aniso-tropy exhibits significant uncertainty.
Résumé : L’interprétation d’essais à charge variable dans des forages tubés est discutée. Ces essais sont communémentutilisés pour évaluer la conductivité hydraulique des barrières d’argile compactée. Ils font généralement partie du pro-gramme d’assurance qualité pendant la construction. Trois méthodes d’interprétation sont abordées. Deux de ces métho-des sont d’emploi courant et ont été le sujet de plusieurs travaux de recherche : celle de Hvorslev et celle des vitesses.Après une discussion mettant en contraste les particularités et les limites de ces deux méthodes, une troisième méthodeest proposée. Celle-ci utilise la technique du meilleur estimateur linéaire non-biaisé pour lisser la courbe théorique dela différence d’élévation de la colonne d’eau en fonction du temps. La comparaison conclut que, pour les argiles com-pactées, la méthode de Hvorslev conduit à des valeurs non fiables parce qu’on y suppose a priori la position d’un ni-veau piézométrique inconnu. Celle des vitesses permet en théorie de trouver ce niveau piézométrique mais l’incertitudestatistique peut être élevée dans le cas des faibles conductivités hydrauliques. La méthode Z–t proposée est moins sen-sible aux incertitudes de mesure. Une méthode d’interprétation pour les étapes 1 et 2 de l’essai à deux étapes en forage(« two-stage borehole ») est proposée. Elle permet de calculer l’anisotropie de la barrière et la conductivité hydrauliqueverticale. Cette anisotropie ne peut pas être connue avec précision étant données les incertitudes de mesure et la faibledifférence entre les géométries des étapes I et II de l’essai.
Mots clés : barrière hydraulique d’argile, couverture en argile, conductivité hydraulique, perméabilité, essai in situ, ani-sotropie, interprétation.
Chiasson 90
Introduction
In the early 1980s, a number of regulatory bodies adoptedprograms to manage all domestic solid wastes producedwithin a geographically defined region. Such programs havepermitted the construction of controlled regional sanitarylandfills that have gradually replaced local unsupervisedopen-pit dumps. Wastes are now stored in isolation cells that
restrict outflow of contaminated liquids (mainly contami-nated water). Cell designs typically include a hydraulic bar-rier layer overlaid by a leachate drainage system. Duringoperations, wastes are gradually mounded in the cell. Oncethe design capacity is reached, the cell is capped with a sec-ond hydraulic barrier (cover), thus encapsulating and isolat-ing the waste.
Modern regulations for base-layer hydraulic barriers inlandfills have the goal of protecting groundwater (Chen andLiew 2003). Typically, the regulation requires a minimumbreakthrough criterion, such as 25 years under a unit hydrau-lic gradient flow. In the case of the U.S. Environmental Pro-tection Agency (U.S. EPA 1991), the regulations providetwo design options for owners of municipal solid-wastelandfills. The first option is a composite liner system, whose
Received 16 December 2002. Accepted 24 June 2004.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 15 February 2005.
P. Chiasson. Secteur de génie civil, Faculté d’ingénierie,Université de Moncton, Moncton, NB E1A 3E9, Canada(e-mail: [email protected]).
minimum requirements include a 0.60 m thick compactedclay liner (CCL) with a hydraulic conductivity no greaterthan 1 × 10–9 m/s. Quality control for liner construction hasled to the development and in some instances standardiza-tion of a number of field hydraulic conductivity testingmethods. Sealed double-ring infiltrometer (SDRI) tests,American Society for Testing and Materials (ASTM) testmethod D5093 (ASTM 2004a), and lysimeter tests are ex-amples. These permit one to test CCL compliance by mea-suring flow through both large- and small-scale defects, suchas bad lift-to-lift bonding and insufficient destruction ofclods and interclod macropores. Unfortunately, they requirelong testing times, which can exceed 3 weeks (Trautweinand Boutwell 1994). This renders these tests impractical forconstruction quality assurance (CQA). They are used, rather,on test pads to demonstrate compliance during the designphase. In this design philosophy, field tests serve to establishconstruction procedures that best minimize defects in theliner. These procedures are then used to construct the actualliner. During construction, the procedures adopted must befollowed exactly and controlled through a well-conceivedCQA program.
Time being an important factor in regions where the con-struction season is rather short, a need exists for faster hy-draulic conductivity tests. One of these tests, which aims todecrease testing times, is the two-stage borehole (TSB) test(test method D6391; ASTM 2004b). This test combines twowell-established borehole test methods: end of casing andinjection through an extended borehole chamber (CAN–BNQ 1988a, 1998b; AFNOR 1992). Trautwein and Bout-well (1994) indicated that this test is more than three timesfaster than SDRI.
Trautwein and Boutwell (1994) proposed special shapefactors for stage I (end-of-casing test) and stage II (extendedborehole). These (as advanced by the authors) could be com-bined and solved for horizontal and vertical components ofhydraulic conductivity. Chapuis (1999) demonstrated theseshape factors to be incorrect. Furthermore, Chapuis doubtedwhether the anisotropy ratio, kh/kv, could be extracted fromsuch a procedure. Chapuis gave two arguments to supporthis affirmation: the two stages have nearly spherical (not el-liptical) equipotentials; and smearing effects introduce highuncertainty into this ratio. Also worthwhile pointing out isan assumption implicitly made by Trautwein and Boutwell(1994): the liner is homogeneous in k. Liners are built froma succession of lifts that may vary in hydraulic conductivityfor reasons such as variations in dry density and water con-tent, soil gradation, and clay mineral content. Cassan (2000)pointed out that the hypothesis of homogeneity in k is legiti-mate at a testing scale where water flows through all repre-sentative soil layers. This does not appear to be the case in aTSB test.
Even with a good CQA program, the importance of hy-draulic conductivity field tests must be emphasized. It isthrough such tests that bad or faulty construction practicewas identified in the past. Early liner construction oftenshowed large discrepancies between field and laboratoryhydraulic conductivity values (Daniel 1984; Elsbury et al.1990). Although it was well known that hydraulic conductiv-ity measurements on samples compacted on the dry side are
many orders of magnitude more permeable than samplescompacted on the wet side (Bjerrum and Huder 1957;Lambe 1958a, 1958b; Mitchell et al. 1965), CQA programsdid not use acceptance criteria based on this widely demon-strated experimental result until quite recently. Work byBenson and Boutwell (1992), Leroueil et al. (1992), Danieland Koerner (1995), Daniel (1998), Benson et al. (1999),and many others gradually permitted the evolution of a CQAbased on compaction control points falling within a zone ofacceptance. This zone of acceptance is defined by a plot ofunit dry mass versus water content (the same as for a com-paction curve). On the basis of optimum conditions for hy-draulic conductivity, the zone is delimited on its left side(lowest water contents) by the line of optimums and on itsright side (highest water contents) by the saturation curve.Other criteria, such as shear strength and the potential togenerate shrinkage cracks, may be added to circumscribe anoverall acceptance zone (Daniel and Wu 1993).
Even when compaction control points fall on the wet sidewithin the overall zone of acceptance, there is still potentialfor noncompliance. Other defects can yield a CCL that failshydraulic conductivity requirements because of bad bondingbetween lifts; insufficient water content blending in the linermaterial during compaction; or environmental factors suchas desiccation and freezing. Thus, field hydraulic conductiv-ity tests are still warranted and should be part of a goodCQA program.
A safer practice would be to include such controls on theCCL during and after final construction. For practicalreasons, controls must be fast and reliable. Otherwise, con-struction may be severely delayed, creating excessive expen-ditures. TSB tests fall into this category. They are fast, withresults obtained in 24 h. They also have minimal destructiveimpact on the CCL, thus minimizing subsequent repaircosts.
A final performance check of the liner can be donethrough a full-scale test for total leakage versus water loadwithin the cell (Chapuis 2002). Such a test has the merit ofdetecting defects that have been missed by field tests or theCQA program. Performing the test for a long duration(greater than 3 weeks) can also help evaluate whether theliner is prone to suffosion (internal erosion). Although sucha test may appear as overkill, detection of defects before theliner enters into service makes it less costly to repair than itwould be once the site is in operation.
This paper reviews data interpretation of cased borehole,specifically TSB, tests. Two known methods of interpreta-tion are reviewed: the Hvorslev method and the Chapuis ve-locity method. A third method is also introduced. All threemethods are compared for their precision in the statisticalsense when field hydraulic conductivity, k, is estimated. Rec-ommendations are proposed for interpreting such measure-ments.
Site description and testing program
A number of cased borehole falling-head hydraulic con-ductivity tests were performed at the Red Pine RegionalSanitary Landfill. The landfill is located in northeastern NewBrunswick (Canada), 20 km south of Bathurst. It is part of a
government program adopted in 1985 and designed to man-age solid waste in a safe and controlled manner. The sitecurrently collects and stores 86 000 t/year of waste.
The cell design at Red Pine includes a base liner of com-pacted clay (1.4 m) overlaid by a high-density polyethylenegeomembrane. To meet the New Brunswick Departmentof the Environment breakthrough criteria, this clay layermust have a hydraulic conductivity of no more than 1.4 ×10–9 m/s.
The construction material, clayey red till and gray transi-tion till that is readily available on the site, has the potentialto meet requirements. Chiasson et al. (1998) reported labora-tory hydraulic conductivity values on specimens compactedat optimum water content under modified energy. Both ma-terials and their mixtures were found to have values nogreater than 0.29 × 10–9 m/s. Mineralogical studies showedhigh clay mineral content for both native soils, thus explain-ing their potential to attain low hydraulic conductivity.Means and standard deviation (SD) for plastic limits, liquidlimits, and plasticity indexes of the two materials are, re-spectively, 18.9% (1.1%), 30.0% (2.5%), and 11.1% (2.2%).
During the course of landfill operations, a number of testpads were built and field tested. Data presented in this paperare for one test pad built in 1995, where an extensive testingprogram was performed. Construction crews and practicesfor the pad were the same as for the real storage cell beingbuilt at that time. The 1995 test pad was 1 m thick and builtof clayey till only. Very little variation in water content wasreported during construction. Compaction controls yieldedan average maximum dry density of 2.015 Mg/m3, with av-erage water content of 10%, corresponding to 80% satura-tion. Following construction of the test pad, an SDRI testwas performed (Jacques Whitford Ltd. 1995). Total durationof infiltration was 25 days. Shelby tube samples, extractedbefore SDRI testing, showed an average water content of10%, with little variation throughout the sampled depth of0–300 mm. Tensiometers were also installed within the outerring of the SDRI at depths of 50, 100, and 150 mm. Brieflyafter installation and before SDRI testing, suction readingsrapidly stabilized at 80 kPa. Following initiation of SDRI in-filtration, suction readings gradually declined to stabilize at5 kPa after 8, 15, and 21 days at the three respective depths.This indicates that compacted clay had attained satiation atthese depths at those times of testing. Following removal ofthe SDRI apparatus, water content of 14% was measured upto a depth of 220 mm. Cased borehole tests reported in thispaper were performed on this pad the next year, in July1996. Although water content was not measured when thesetests were performed, the 1 year resting period must have fa-voured dissipation of suction values throughout the thicknessof the liner, yielding satiated compacted clay.
Cased borehole tests performed on the 1995 test pad wereof the end-of-casing type. Acrylonitrile butadiene styrenepipes had an internal diameter of 102 mm (4 in.). The testingcampaign used medium sodium bentonite chips (9.5 mm)and sodium bentonite grout to seal the annular space be-tween borehole sidewall and pipe. As reported by ASTM(2004b), Chapuis (1998), and Trautwein and Boutwell(1994), wall smearing of the injection zone may influenceresults. To prevent this problem, the soil surface at the end
of casing was inspected and manually cleaned accordingto ASTM D6391 recommended practice (ASTM 2004b).Finally, permeability tests in compacted clay were precededby a wetting period of 24–48 h.
Interpretation of falling-head sealedborehole tests
Chapuis (1998) indicated that when the deformations ofthe soil can be neglected, falling-head tests are governed bythe Laplace equation. Its solutions, the harmonic functions,have several properties. One of them relates the flux in thesoil (Qsoil) to the flow into the pipe (Qinj) through a mass-balance equation,
[1] Qinj = Qsoil = ckH
where c is a shape factor that depends on the geometry ofthe injection zone and on the hydraulic boundaries of theproblem; H is the applied hydraulic head difference; and k isthe hydraulic conductivity. This equation is the starting pointof both the Hvorslev and the Chapuis methods. Anotherequation is the starting point of another method for caseswhere soil deformation is assumed to be elastic and notnegligible (Cooper et al. 1967). However, the mathematicalmodel of this method does not correspond to the physicalmodel. This is according to mathematical, physical, and nu-merical proofs by Chapuis (1998) and experimental proofsby Chapuis and Chenaf (2002). According to the equationsof Chapuis (1998), the effect of soil deformation can be ne-glected when the soil is an aquifer or an overconsolidatedaquitard. Soil deformation is no longer negligible for com-pressible aquitards when they are tested by using either afalling-head test with a very small injection pipe or a pulsetest between packers. Chapuis and Cazaux (2002) suggestedmethods for handling the instantaneous (elastic) and delayeddeformations in such cases. Because this paper deals onlywith compacted clays, which are overconsolidated aquitards,the use of eq. [1] is thus justified. In a falling-head test, Qinjis the flow through the inflow pipe (often a standpipe con-nected to the borehole casing) of internal cross section Sinj.
[2] Q SHt
inj injdd
= −
where t is time. Equations [1] and [2] yield
[3]dd inj
Ht
ckHS
= −
Rearranging gives
[4]d
dinj
HH
ckS
t= −
Integrating leads to Hvorslev’s solution (1951):
[5] ln ( ) ( )HH
kc
St t kC t t1
21 2 1 2
⎛
⎝⎜
⎞
⎠⎟ = − − = − −
inj
where H1 and H2 are, respectively, the applied differences intotal head at times t1 and t2; and C = c/Sinj. If the falling-
head test is made with a standpipe of internal diameter d,then
[6] Sd
inj = π 2
4
Shape factors
The following shape factors are best suited for the prob-lem at hand (Chapuis 1999). For the end-of-casing test, theshape factor C is
[7] CD
d= 11
2π
For the Lefranc test, which is with a cylindrical injectionzone, the sphere formula is recommended when 1 ≤ L/D ≤ 8:
[8] c DLD
= +214
π
which gives
[9] Cc
SD
dLDinj
= = +8 142
when 1 ≤ L /D ≤ 8. These latter two equations are, accordingto Chapuis (1999), accurate to a relative error of 10%. Moreaccurate shape factors can be obtained from modified equa-tions that take into account the upper and lower boundariesof the barrier (Chapuis 1989).
Hvorslev method of interpretation
In the Hvorslev method, data are plotted on a semiloggraph with ln [H(t = 0)/H(tj)] on the ordinate and time t onthe abscissa. According to eq. [5], this should theoreticallygive a straight line. Let m be the slope of this line, such thatm = [ln (H1/H2)]/(t1 – t2). Then, from eq. [5],
[10] m = –kC
and hence
[11] k = –m/C
In the case of TSB tests in a CCL, the true difference inhydraulic head at time t is unknown. What is measurable attime t is elevation Z of the water column from a set refer-ence level. Typically, this is chosen as the surface of thebarrier. The unknown component is the height of the piezo-metric level (PL) of the soil around the injection zone fromthe set reference (Fig. 1). The correct difference in head attime t is
[12] H(t) = Z(t) – Ho
where Ho is the unknown height of the soil PL from the setreference level (positive, if over; negative, if under). This no-tation is in accordance with that of a number of authors(Schneebeli 1954; Chapuis et al. 1981; CAN–BNQ 1988a,1988b) who use Ho to symbolize this unknown height. Thisshould not be confused with the definition employed in theHvorslev method (1951), where Ho is the initial head differ-ence.
In normal practice, the operator is too often unaware ofthe unknown height Ho. The operator thus sets the PL at anassumed level and sets this level as the datum. By doing this,the operator is not plotting ln[H(t = 0)/H(tj)] but ratherln[Z(t = t0)/Z(tj)] versus tj, which from eq. [5] gives
[13] ln( )( )
( ) ln ( )Z tZ t
kC t t Z tj
j0
0 0
⎡
⎣⎢⎢
⎤
⎦⎥⎥
= − − +
− − − − −ln[ ( ) ( )]( )Z t H kC t tj0 1 0
o e
where t0 and Z(t0) are initial readings of time and watercolumn elevation; and tj and Z(tj) are the jth readings.
This plot gives a straight line only when the datum is setat the same level as the correct PL of the soil (in otherwords, when the hypothesis Ho = 0 is correct). When Ho ≠ 0,the plot of ln [Z(t0)/Z(tj)] versus tj yields a more or less pro-nounced upward or downward curve (Fig. 2). Furthermore,inspection of eq. [13] shows that when Ho ≠ 0, eq. [13] can-not be simplified to eq. [5], and therefore the relationshipbetween hydraulic conductivity and the slope of the plot asexpressed by eq. [11] is no longer valid. Anyone who is notaware of this will naturally assume that eq. [11] still applies.This would lead to an erroneous interpretation of the plotand erroneous hydraulic conductivity values. Because soilscomposing a CCL are in an unsaturated state, a PL belowthe top of the liner should be expected. In the author’s expe-rience, the correct PL for a fairly humid climate is often
Fig. 1. Typical setup of a cased borehole test in a compactedclay liner.
Fig. 2. Semilog plot of falling-head test. Datum and assumedpiezometric level at top of compacted clay liner (see data in Ta-ble 1).
found somewhere between the top and the bottom of theliner (see also Chapuis 1999). In an illustrative example (Ta-ble 1, Fig. 2), the datum is set at the top of a 1 m CCL, andthe soil PL is assumed to be at the same level. This leads tothe erroneous conclusion that hydraulic conductivity in-creases with time (Table 1, Fig. 3: PL assumed to be at thetop of the CCL). If the datum and assumed PL are set at thebottom of the liner (add 100 cm to Z in Table 1), the hydrau-lic conductivity decreases slightly with time, thus suggestinga relatively constant value of k = 1.4 × 10–10 m/s (Fig. 3: PLassumed to be at the bottom). Finally, choosing the correctPL, at 44.55 cm below the top of the liner, yields a constantk = 3.51 × 10–10 m/s throughout the duration of the test.Three different results are found for three different assumedPL levels. Test interpretation through the Hvorslev methodyields a hydraulic conductivity, k, that is a function of theassumed PL. Use of this method does not appear to be rec-ommendable.
The velocity (Chapuis) method
According to eq. [3], plotting of falling-head velocity as afunction of applied head difference should yield a straightline. Because the applied head difference is usually un-known, combining eqs. [3] and [12] yields
[14]dd
oZt
Ck Z H= − −( )
and rearranging gives
[15] ZZt Ck
H= − +dd
o1
Note that in a falling-head test, the velocity, dZ/dt, is neg-ative, as a result of the downward direction of flow. Accord-ing to this last equation, the plot should yield a straight linefor whatever set datum. The intercept Ho is the error, or theheight separating the datum and the correct PL of the soil(Fig. 1). The slope of the plot is mv = –1/Ck. Therefore,
[16] k = –1/Cmv
The slope, mv , is not a function of the unknown PL of thesoil. Thus, a hydraulic conductivity computed from a plot ofheight Z as a function of velocity dZ/dt is independent of thePL of the soil.
In practice, the mean elevation, Zm, during time increment∆t is plotted as a function of average velocity, v = ∆Z/∆t. Aplot of data from Table 1 gives a straight line (Fig. 4) with aslope mv = 233.7 min and an intercept Ho = –44.55 cm.Computing k gives 3.51 × 10–10 m/s. The Hvorslev methodgives this same value if the datum is set 44.55 cm below thetop of the liner.
To resume, the velocity method does not depend on theset datum (or assumed PL); furthermore, it yields the correct
Note: Datum and assumed piezometric level of soil at top of compacted clay liner (Lefranc type test, shape factorC = 2032 cm–1).
Table 1. Falling-head test in a cased borehole on the 1995 test pad.
Fig. 3. Hydraulic conductivity computed for assumed piezometriclevel at top, bottom, and correct level (44.55 cm below top) withHvorslev’s method (see data in Table 1).
Fig. 4. Velocity method plot of a cased borehole test (see data inTable 1).
PL of the soil. Unfortunately, this method has its pitfalls. In-accuracies in measurements can generate larger inaccuraciesin computed velocities. This is due to the problem of doubleinaccuracy in derivatives, with the potential to produce con-siderable scatter in the velocity plot (see data in Table 2 andFig. 5). Least squares yields an intercept Ho = 100.97 cmand a slope mv = 29.59 min, which corresponds to k = 2.4 ×10–9 m/s. Applying the Hvorslev method to the falling eleva-tion data of Table 2, with a new datum set at 100.97 cmabove the original one, yields a curved plot in the semiloggraph (Fig. 6). This curvature underlines an inappropriatecorrection of the PL, as demonstrated earlier by eq. [13]. Af-ter inspection of the velocity plot and the semilog plot (seeFigs. 5 and 6), one would not put much confidence in thevelocity method value of k or in the test itself. The velocityplot has a mediocre fit (R2 = 0.1832), whereas the semilogplot is clearly curved downward. The inaccuracy amplifiedin the velocity plot of Fig. 5 may lead one to classify thistest as poor or defective, and the temptation may be to rejectit. Results such as this motivated the development of themethod described in the following section.
Proposed interpretation method: Z–t
The objective is to develop a method less sensitive tomeasurement inaccuracy. Because relative inaccuracies onwater column elevation Z and time t are generally small(<1%), a plot of these data should display little scatteraround the trend. Rearranging eq. [5] results in H(t) becom-ing
[17] H(t) = H(0)e–kCt
Setting
[18] a = kC
and combining with eq. [12] yields
[19] Z(t) = [Z(0) – Ho]e–at + Ho
Because the expression in brackets corresponds to H(0),the following is preferred:
[20] Z(t) = H(0)e–at + Ho
where Z(t) is water column elevation as a function of time t;and H(0) is the (true and unknown) hydraulic head differ-ence at t = 0. A best unbiased estimator is used to evaluatethe unknown parameters: H(0), Ho, and a. The residual be-tween the estimator of the function Z*(t) and the measure-ment Z(tj) is
[21] ε j j jZ Z t= − *( )
The solution for Z*(t) is obtained by minimizing the sumof squared residuals,
Note: Lefranc type test, shape factor C = 2353 cm–1.
Table 2. Falling-head test in a piezometer installedin Champlain clay below the weathered surfacelayer and phreatic line.
Fig. 5. Velocity method plot for a falling-head test in a piezo-meter installed in Champlain clay (Lefranc test; see data in Ta-ble 2).
Fig. 6. Semilog plot for a falling-head test in a piezometer in-stalled in Champlain clay (Lefranc test; see data in Table 2).
where n is the number of measurements Z(tj). The complex-ity of eq. [20] does not permit an analytical solution to thisproblem. Numerical optimization techniques are thereforerequired. These are readily available in most modern spread-sheets. The solution for hydraulic conductivity, k, will be
[24] k = a/C
Deriving eq. [20] as a function of time t yields the slopeof the velocity graph:
[25] mv = dZ/dt = –1/a = –1/kC
Plotting the data of Table 2 and applying the method yieldsH(0) = 71.79 cm, Ho = 51.15 cm, and a = 0.00730 min–1
(Fig. 7). Equation [24] gives k = 5.2 × 10–10 m/s. As expected,the statistical scatter around the trend is low. The hydraulicconductivity from this method is smaller by a factor of 4.6than that obtained from the velocity method. The goodness offit illustrated in Fig. 7 gives more confidence in this hydrau-lic conductivity value and suggests that the test is of goodquality. Furthermore, a regression line through a semilogplot of corrected falling elevation data from Table 2, with anew datum set 51.15 cm above the original reference, yieldsa correlation coefficient, R2, of 0.9996 (Fig. 8). This impliesthat the estimated Ho obtained by the proposed method is abetter estimate of the true correction to be applied to thedata.
Comparing methods
For comparison of statistical robustness, the three studiedmethods—the velocity method, the Hvorslev method cor-rected by Ho as estimated from the velocity method, and theHvorslev method corrected by Ho as estimated by the Z–tmethod—and the proposed Z–t method were applied on 19falling-head test trials. These tests were performed on a CCLtest pad built in 1995 on the Red Pine site. The clay wascompacted at 94.5% of the modified Proctor dry density. All19 test trials were performed in borehole No. 5 and used thesame end-of-casing geometry. The velocity and the Z–t me-thods of interpretation yield values of the same order ofmagnitude, although the average hydraulic conductivity of 1.9× 10–9 m/s for the proposed method is lower than the 2.4 ×10–9 m/s obtained from the velocity approach. The velocitymethod also yields higher statistical scatter than the Z–tmethod (Fig. 9). With the exception of one test, the proposedmethod systematically yields lower hydraulic conductivityvalues (Figs. 9 and 10). It also gives the lowest SD for k,with 0.6 × 10–9 m/s versus 0.9 × 10–9 m/s. Thus, the preci-sion of computed k values is 33% better with the proposedZ–t method.
In the case of the velocity method, the standard error onthe slope permits one to compute the 95% confidence inter-val for k (Neter et al. 1989). As a basis of comparison forboth methods, the coefficient of determination (R2) com-puted for each trial run is used. Both quantities are used asindexes to characterize statistical dispersion of each testtrial. Test trials with low data scatter have narrower errorbars than tests with higher scatter (Fig. 10).
In tests with low scatter, similar hydraulic conductivityvalues are computed whatever the method used. This is nottrue for tests with high data scatter. In such tests, the veloc-
Fig. 7. Elevation of water column as a function of time forfalling-head test data in a piezometer installed in Champlain clay(Lefranc test, see data in Table 2).
Fig. 8. Semilog plot of corrected falling-head test data in apiezometer installed in Champlain clay. Data corrected by Z–tmethod (Lefranc test; see data in Table 2).
Fig. 9. Cumulative frequency of hydraulic conductivity for 19tests as computed with the velocity method; the Hvorslev methodcorrected by Ho as estimated by the velocity method; the pro-posed Z–t method; and the Hvorslev method corrected by Ho asestimated by the Z–t method. All tests were performed in bore-hole No. 5, using the same end-of-casing geometry, on a com-pacted clay liner built in 1995.
ity and Z–t methods of interpretation yield different hydrau-lic conductivity values. For example, test trial No. 19 yields3.0 × 10–9 m/s with the velocity method, whereas the pro-posed Z–t method gives 1.1 × 10–9 m/s. The latter value is2.8 times smaller. As shown in Fig. 11, hydraulic conductiv-ity computed with the velocity method tends to increasewith decreasing quality of fit (decreasing R2). In the case ofthe proposed method, hydraulic conductivity is not sensitiveto random inaccuracies in measurements, because no signifi-cant correlation is found between computed hydraulic con-ductivity and quality of fit (R2 = 0.064 in Fig. 12). Note alsothat the proposed method yields very high coefficients of de-termination. This indicates that the curve fits the data well(such as illustrated in Fig. 7) and explains well the interrela-tionship between time t and falling water column elevationZ. The same data, when interpreted with the velocitymethod, yield variable coefficients of determination, withmany tests appearing to be of poor quality. Meanwhile, veryhigh coefficients of determination are systematically ob-tained for the proposed method (Table 3). Thus, the pro-
posed method is statistically more robust. This is evidentwhen comparing the goodness of fit of the velocity plots(Fig. 5) with that of the Z–t plots (Fig. 7). With the velocitymethod, small inaccuracies in measurements generate ratherlarge relative errors in the velocity values. Scatter is ampli-fied in the velocity plot, from which follows the high uncer-tainty of k.
The Hvorslev method does not prove to be very useful.When the correct head difference of eq. [12] uses Ho com-puted with the velocity method, the Hvorslev method yieldsthe same k value as obtained from the velocity approach(Fig. 9). If the same is done using Ho obtained from the pro-posed Z–t method, the Hvorslev method then gives the sameresult as the proposed method (Fig. 9). Thus, the Hvorslevmethod will agree with the method with which the Ho cor-rection is obtained, even though the velocity method and theproposed method yield different hydraulic conductivity val-ues.
Evaluating the anisotropy of hydraulic conductivitySoils commonly display anisotropy in hydraulic conduc-
tivity, α = kh/kv ≠ 1. In a stratified soil, this ratio will alwaysbe greater than 1. According to eq. [1] and Cassan (2000),the flow from an injection zone into a transformed mediumis
[26] QQ
Q c k Hinjinj
soil′ = = ′ = ′ ′ ′α
where c′ is the shape factor of the injection zone for thetransformed medium; k′ = kv; and H′ = H.
In the shape factor c′, ′ =D D/( )./α1 2 The shape factor c′ isa function of the injection zone aspect ratio, λ′, in the trans-formed medium, where
[27] λ λ α α′ = = LD
Depending on the ratio of anisotropy, the transformed ra-tio can be significantly greater than the true physical aspectratio. Therefore, the shape factor function for the trans-formed medium can be different from that for the physical
Fig. 10. Hydraulic conductivity computed with the proposed Z–tmethod versus that computed with the velocity method with er-ror bars (end-of-casing tests performed in borehole No. 5, 1995test pad).
Fig. 11. Hydraulic conductivity computed with the velocitymethod as a function of the coefficient of determination for 19test trials (end-of-casing tests performed in borehole No. 5, 1995test pad).
Fig. 12. Hydraulic conductivity computed with the Z–t methodas a function of the coefficient of determination for 19 test trials(end-of-casing tests performed in borehole No. 5, 1995 test pad).
medium. Following development of the velocity method,eq. [15] can be rewritten as
[28] ZdZdt C k
H= −′ ′
+1α e
where C c S′ = ′/ inj.The velocity plot of average elevation, Zm, during a time
increment as a function of dZ/dt has the slope
[29] mk C
v = −′
1α v
The vertical hydraulic conductivity of the medium is then
[30] kC mv
v = −′1
α
The shape factor C′ of the transformed injection chamberis a function of the unknown anisotropy, α = kh/kv. To solveunknowns α and kv, Trautwein and Boutwell (1994) sug-gested performing borehole tests of varying geometry. Thetest is first performed with the casing flush with the holebottom (end-of-casing test). This is stage I of the test. StageII infiltration is measured with an extended borehole. Duringstage I,
[31] kC m
v = −′
1
1 1α
and during stage II,
[32] kC m
v = −′1
2α 2
where C1′ and C2′ are, respectively, shape factors for stage Iand stage II; and slopes m1 and m2 are obtained, respectively,
from velocity plots (or Z–t plots and eq. [25]) for tests per-formed in stage I and stage II. Computing the quotient ofeqs. [31] and [32] gives
[33]CC
mm
2
1
1
2
′′
=
Both C1′ and C2′ are functions of the ratio of anisotropy,α = kh/kv. During stage I of the TSB test described byTrautwein and Boutwell (1994), shape factor C1′ is, accord-ing to eq. [7],
[34] CDd
Dd
1 2 2
11 1 11′ = ′ =π α π
Because of the limited thickness of the CCL, injectionzones for stage II of TSB tests rarely extend beyond one di-ameter in length. Generally, the shape factor for stage II willcorrespond to eq. [9], giving
[35] CD
dL
D2 2
1 8 14
′ = +α
α
when 1 < (Lα1/2)/D < 8. Note that (Lα1/2)/D is the trans-formed aspect ratio λ 2′ , and the shape factor C is as definedin eq. [9]. Equation [33], combined with shape factors forstage I and II, then gives
Note: All 19 test trials were performed in the same borehole on the 1995 test pad.
Table 3. Hydraulic conductivity (k), coefficient of determination (R2), and relative error onk (εk/k) as computed with velocity and Z–t methods.
Theoretically, eqs. [36] and [37] give the ratio of aniso-tropy from measurements for stages I and II of a TSB test,and eq. [31] or eq. [32] permits one to compute the verticalhydraulic conductivity, kv. The ratio of anisotropy as a func-tion of the slope ratio for a typical TSB test, where λ′ iswithin the range 1–8, is illustrated in Fig. 13. From a practi-cal point of view, inaccuracies in measurement cause highuncertainty for anisotropy. An excellent velocity plot, suchas illustrated in Fig. 4, gives 15% relative error on the slope(based on the 95% confidence interval on slope value; Neteret al. 1989). Figure 13 shows a confidence interval whererelative errors on slopes m1 and m2 are both 10%. This confi-dence interval for the ratio of anisotropy spans a wide range.For example, a TSB test in which the ratio for stages I and II(m1/m2) is 4.22 gives a ratio of anisotropy (α) of 10, with aconfidence interval extending from 3.7 to 22. This is inagreement with observations made by Chapuis (1999) and asearlier described in the introduction. Therefore, these equa-tions should be used with caution.
Conclusion
Three methods of interpretation for borehole falling-headtests were reviewed. Hydraulic conductivity values com-puted with the Hvorslev method were found to strongly de-pend on the correct PL of the soil. Because the PL is usuallyunknown, a level must be assumed. Depending on the eleva-tion at which PL is set, the method can yield an increasing k,a decreasing k, or a k that appears constant during the test.
With the velocity method, the unknown PL of the soil isnot needed for computing hydraulic conductivity. Therefore,an error in the assumed PL has no consequence for computa-tions. When scatter in the velocity plot is low, the methodyields good hydraulic conductivity values (these are found toagree with the proposed method). Unfortunately, even smallinaccuracies in measurements create large inaccuracies incomputed velocities, particularly when testing materials with
low hydraulic conductivity. Furthermore, computations willyield higher k values as scatter in the velocity plot increases.
In a velocity plot, average water column elevation (Zm)and falling-head velocity, v = dZ/dt, are both dependent vari-ables. Least-squares estimation is theoretically based on onedependent variable being a function of another that is inde-pendent. With low scatter, the choice of one or the other asthe dependent variable has little consequence. With highscatter, k values depend on which variable is set as the inde-pendent variable and which is set as the dependent variable.The data in Table 2 yield k = 2.4 × 10–9 m/s when y = v andx = Zm. The same data produce k = 4.4 × 10–10 m/s if y = Zmand x = v. The k value being a function of which variable isset as x and which is set as y is attributable to a hypothesisbehind the least-squares method: x is an independent vari-able with no inaccuracies, and y is a dependent variable withunknown random inaccuracies in measurements. Unfortu-nately, both velocity, v, and average elevation, Zm, are depen-dent variables, with both having some random inaccuracies.Hence, they do not respect, in the strict sense, the hypothesisbehind the least-squares method. Where inaccuracies aresmall, they will respect the hypothesis in a relaxed sense. Insuch cases, they will yield k values comparable with those ofthe Z–t method and will not be sensitive to which variable isset as x and which is set as y (data in Table 1, for example).These observations lead us to conclude that the velocitymethod is not statistically robust. It should be used with cau-tion when scatter in the velocity plot is observed.
The proposed Z–t method is found to be statistically morerobust. It is independent of random inaccuracies in measure-ments (at least in the orders of magnitude found in routinetests). With this method, the intrinsic hypothesis mentionedearlier for least-squares estimation is respected. Water col-umn elevation (Z) and time (t) are, respectively, true depend-ent and independent variables. Plots also typically displaylittle data scatter, which clearly shows the theoretical interre-lationship of t and falling water column elevation (Z) ofeq. [20].
Theoretically, the Hvorslev method should yield a correctk when the correct PL of the soil is used for computing headdifference. In practice, inaccuracies in measurements yieldhigh uncertainty in the estimation of the correct PL. This isdue to an important trend extrapolation to obtain Ho in boththe proposed and the velocity methods. The Hvorslevmethod will agree with the method with which the PL is es-timated, even though the velocity and proposed methodsyield different hydraulic conductivity values. In light of this,the direct use of the Hvorslev method without questioningthe PL value is not recommended.
An interpretation method for stages I and II of TSB testswas also proposed. It yields the anisotropy and vertical hy-draulic conductivity of the liner. As a result of inaccuraciesin measurements and limited difference in geometry betweenstages I and II, computed anisotropy has high uncertainty.
Acknowledgements
This research on the interpretation of falling-head bore-hole tests was supported by the Nepisiguit-Chaleur SolidWaste Commission and in part by the Natural Sciences and
Fig. 13. Hydraulic conductivity ratio of anisotropy as a functionof slopes m1 and m2 obtained from velocity plots of stages I andII. Relation valid when, for stage II, 1 < λ′ < 8. Dashed linesare confidence intervals obtained when relative errors on slopesm1 and m2 are both 10%.
Engineering Research Council of Canada and by the Facultédes études supérieures et de la recherche de l’Université deMoncton. The author is grateful for the assistance offered byMr. Raymond Bryar, manager of the Commission, and byMr. Marc Antoine Caissie, engineer with the Roy Consul-tants Group. Field testing performed by Benjamin Chiasson,Rémi Godin, and Wamytan Rezza is also acknowledged.The author wishes to thank Mr. Camille Vautour, technician,and Mrs. Jolaine Landry, laboratory engineer, for their tech-nical assistance throughout this research program. Specialthanks are due to Dr. Robert Chapuis of the École poly-technique de Montréal for his constructive comments andhelpful advice during the preparation of this paper.
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