Abstract: An alternative procedure to determine the hydraulic conductivity function (k-function) based on relationships between saturated
hydraulic conductivity and void ratio and between air-entry value and void ratio is proposed. The procedure was applied to determine the k-function
of deinking by-products, a highly compressible industrial by-product that is used as alternative material in geoenvironmental applications. The
validity of the procedure is verified by comparing the k-function of a compressible soil obtained based on the proposed procedure (using published
experimental data) with experimentally determined unsaturated hydraulic conductivities for the same soil

Cabral, Alexandre

Parent, Serge-Étienne

English

Savoirs UdeS

Geotechnical Testing Journal, Vol. 27, No. 6Paper ID GTJ11854Available online at: www.astm.orgTECHNICAL NOTESerge-Étienne Parent,1 Alexandre Cabral,2 Eduardo Dell’Avanzi,3 and Jorge G. Zornberg4Determination of the Hydraulic ConductivityFunction of a Highly Compressible MaterialBased on Tests with Saturated SamplesABSTRACT: An alternative procedure to determine the hydraulic conductivity function (k-function) based on relationships between saturatedhydraulic conductivity and void ratio and between air-entry value and void ratio is proposed. The procedure was applied to determine the k-functionof deinking by-products, a highly compressible industrial by-product that is used as alternative material in geoenvironmental applications. Thevalidity of the procedure is verified by comparing the k-function of a compressible soil obtained based on the proposed procedure (using publishedexperimental data) with experimentally determined unsaturated hydraulic conductivities for the same soil.KEYWORDS: hydraulic conductivity function, compressible soils, suction, characteristic curve, deinking by-productsIntroductionBecause of experimental difficulties for direct measurement ofthe unsaturated hydraulic conductivity of porous materials, currentpractice involves estimation from water retention curve (WRC)models (Mualem 1976; Fredlund and Xing 1994; Huang et al. 1998;Fredlund 2002). Consequently, the reliability of the hydraulic con-ductivity function (k-function) depends not only on the accuracyof the WRC, but also on the model used for its indirect determin-ation. This study presents a procedure to determine the k-functionof highly compressible materials, based on hydraulic conductivitytests performed on saturated samples and on suction test data per-formed in order to obtain the air-entry value. Suction tests must bedone so that volume changes can be monitored using methodologiessuch as proposed by Cabral et al. (2004). The validity of the proce-dure is verified by comparing results of actual unsaturated hydraulicconductivity tests (published experimental data for a compressiblesoil) with predictions made by applying the procedure.Materials and MethodsDeinking by-products (DBP) are produced in the early stages ofthe paper recycling process. The main inorganic components of thismaterial are calcite, meta-kaolinite, and rutile, whereas cellulose,hemicellulose, and lignine are the main organic components (Cabralet al. 2002). Laboratory tests have shown that the consolidationbehavior of DBP resembles that of highly organic materials, insofaras it presents a significant secondary consolidation (creep) phase.Received January 2, 2003; accepted for publication May 12, 2004; publishedNovember 2004.1 Ph.D. candidate, Dept. Civil Eng., Univ. de Sherbrooke, Quebec, Canada.2 Professor, Dept. Civil Eng., Univ. de Sherbrooke, Sherbrooke, Quebec,Canada.3 Assistant Professor, Dept. Civil Eng., Federal University of Paraná, Brazil.4 Assistant Professor, Dept. Civil Eng., University of Texas at Austin.Saturated hydraulic conductivity tests performed in oedometersafter each of several consolidation steps show that the saturatedhydraulic conductivity for this material lies in the range of 5 ×10−10 m/s and 5 × 10−9 m/s (Burnotte et al. 2000). The resultsof several tests, reproduced in Fig. 1, show a linear relationshipbetween the logarithm of the saturated hydraulic conductivity andvoid ratio. This relationship can be expressed as follows:ksat = k′sat 10b(e−e′) (1)where ksat is the saturated hydraulic conductivity for a void ratioe, e′ is an arbitrary reference void ratio, k′sat is the saturated hy-draulic conductivity at e′, and b is the slope of the log(ksat) versuse relationship (Fig. 1).Based on experimental data reported in the literature (e.g.,Laliberte et al. 1966), Huang et al. (1998) assumed—and presentedevidence—that the logarithm of the air-entry value (ψaev) is lin-early proportional to the void ratio at the air-entry value (eaev).That is:log(ψaevψ′aev)= ε(eaev − e′aev) (2)where ψaev is the air-entry value for void ratio eaev, ψ′aev is theair-entry value at the arbitrary reference void ratio e′aev, and ε is theslope of the log(ψaev) versus eaev curve.In the present study, the ψaev values were obtained from suctiontests performed with DBP whose main goals were to determine theWRC of this material and the air-entry value. The methodologyemployed, that allows for consideration of volume changes duringtesting, is described in Cabral et al. (2004). In the cases where a0-bar porous stone was employed (i.e., a porous stone with a neg-ligible capacity to prevent air from breaking through it), the ψaevwas considered to be equal to the suction value where air brokethrough the sample. In the cases where 1-bar porous stones wereemployed, the ψaev was considered to be equal to the suction valuewhere significant loss of water was observed, i.e., in the regionCopyright © 2004 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. 12 GEOTECHNICAL TESTING JOURNALFIG. 1—Void ratio as a function of saturated hydraulic conductivity for deinking by-products.FIG. 2—Water retention curves up to the air-entry value for deinking by-products consolidated to different void ratios.of the inflection point determined using the procedure proposed byFredlund and Xing (1994). Figure 2 shows results from various suc-tion tests performed with DBP with different initial void ratios. Theair-entry values were coupled with their corresponding void ratioin order to obtain Fig. 3, that shows that the relationship betweenlog(ψaev) and eaev is approximately linear, which is consistent withEq 2.Conceptual ModelFigure 4 can be utilized to explain the concept behind the pro-posed model. The k-function is described here by two curves. Thefirst (denoted as Curve 1) describes how the saturated hydraulicconductivity varies with suction for a sample of a highly compress-ible material consolidated to an initial void ratio e0 i . Using a voidratio function (e-function; proposed later), it is possible to predictthe void ratio ei for any suction value ψ and, using Eq 1, to predictthe saturated hydraulic conductivity, ksat i , corresponding to ei andψ. Using Eq 2 and the e-function, the air-entry value for Test i,ψaev i , is then determined. The void ratio corresponding to ψaev iis eaev i .The second curve (denoted as Curve 3 in Fig. 4) can also beestimated based on saturated hydraulic conductivity tests and theWRC. For the same Test i as above, consolidated to an initial voidratio e0 i , when suction exceeds ψaev i , the void ratio decreasesto a value eunsat i lower than eaev i . Accordingly, the associatedhydraulic conductivity decreases—as expected—to a value kunsat ilower than ksat i . This new hydraulic conductivity value, kunsat i ,corresponds to the unsaturated hydraulic conductivity for Sample iwhen its void ratio is eunsat i .According to Eq 1, it is possible to obtain a saturated hydraulicconductivity corresponding to a void ratio eunsat i . In this case,another Test ii (Curve 2 in Fig. 4) must be performed using thesame material, but the sample is consolidated to an initial void ratioPARENT ET AL. ON SATURATED SAMPLES 3FIG. 3—Air-entry value as a function of void ratio for deinking by-products.FIG. 4—Saturated hydraulic conductivity as a function of air-entry value for deinking by-products.e0 ii < e0 i . Suction is then applied to bring the sample to a voidratio eaev ii equal to eunsat i (Fig. 4).Given that, as shown schematically in Fig. 5a, various WRCsuperimpose into a single virgin desaturation branch (Toll 1988)(this is equivalent to the consolidation behavior of clayey soils) andthat the slope of the k-function can be calculated from the slopeof the WRC, it may be inferred that k-functions also superimposeonto the same curve. Therefore, the curve obtained by joining thepoints (ψaev; ksat) for tests i, ii, etc., i.e., Curve 3 in Fig. 4, woulddescribe the k-function of the compressible material for suctionsgreater than the air-entry value.Proposed ProcedureIn order to model Curve 1 in Fig. 4, an e-function must first bedefined. The following model is proposed to describe the suction-induced shrinkage behavior of DBP,ei = ec + (e0 i − ec)(1 + e0 ic1ψ)c2 (3)where ei is the void at Suction ψ, and c1 and c2 are fitting para-meters. The parameter ec is the convergence void ratio, i.e., thevoid ratio at which the e versus ψ curves, obtained from suctiontests with various representative samples, converge. The curves andfitting parameters obtained are presented in Fig. 5b. Although thecurve in this figure extends beyond the air-entry value, obtainmente versus ψ curve beyond the air-entry value is not necessary forthe purpose of determining Curve 1 in Fig. 4. This is due to thefact that the value of ec does not affect significantly the value ofei for suctions lower than ψaev i , whether it is determined by aleast square optimization technique or simply imposed based on aknown residual value (as is the case in Fig. 5b).4 GEOTECHNICAL TESTING JOURNALFIG. 5—Water retention curves for a material consolidated to different initial void ratios (a); Void ratio function for deinking by-products (b).By substituting e in Eq 1 by ei obtained from Eq 3, the followingmodel for Curve 1 (or Curve 2) is obtained:ksat i = k′sat 10b(ec+(e0 i−ec)(1+e0 i c1ψ)c2 −e′aev) for ψ < ψaev i(4a)Curve 3 (Fig. 4) is obtained by replacing the parameters e and e′in Eq 1 by, respectively, eaev and e′aev, by isolating eaev in Eq 2and finally by subsituting Eq 2 into Eq 1. Equation 4b is then usedto establish a linear relationship on a log-log scale between thesaturated hydraulic conductivity and ψaev:ksat = k′sat(ψaevψ′aev)b/εfor ψaev > ψaev i (4b)Validation and DiscussionIn order to validate the proposed procedure, Eqs 3 and 4 wereapplied to experimental data published by Huang et al. (1998). Theresults, presented in Fig. 6, are compared to unsaturated hydraulicconductivity measurements, also reported by Huang et al. (1998).Hydraulic conductivity, suction, and void ratio values obtainedfrom flexible-wall permeability tests allowed for the determinationof the parameters in Eq 4a. Parameters ψ′aev and ε were obtainedusing results of pressure-plate cell tests, from which the curvelog(ψaev) versus eaev could in turn be determined.There seems to be a better agreement between rigid-wall hy-draulic conductivity tests (triangles in Fig. 6) and the curve ob-tained by applying Eq 4. Up to 30 kPa of suction, a good agreementalso exists with results from flexible-wall hydraulic conductivitytests (squares in Fig. 6). The all-data best fit, which was determinedby a least-squares minimization approach using actual unsaturatedPARENT ET AL. ON SATURATED SAMPLES 5FIG. 6—Hydraulic conductivity as a function of suction (data from Huang et al. 1998).hydraulic conductivity tests, was compared to the curve obtained byapplying Eq 4. It can be observed in Fig. 6 that the differences be-tween the proposed procedure and the all-data best fit are relativelyminor up to ψaev i and much less than one order of magnitude forsuctions greater than ψaev i . This suggests that the determinationof the k-function can be made based on known relationships be-tween ksat and void ratio, and ψaev and void ratio with reasonableconfidence.ConclusionsA procedure to determine the k-function based on the relation-ships between saturated hydraulic conductivity and void ratio, andbetween air-entry value and void ratio was proposed and applied toa highly compressible material (deinking by-products). A compar-ison between the k-function obtained by applying this procedureto experimental data reported in the literature (for a Saskatchewansilty sand) and actual unsaturated hydraulic conductivity data forthe same silty sand shows a good agreement up to a suction value inthe vicinity of 30 kPa. For greater suctions, a reasonable agreement(less than one order of magnitude) is still obtained.The use of the proposed procedure to determine the k-functionrequires suction and saturated hydraulic conductivity testing onsamples consolidated to different initial void ratios. However, thesetests are more expeditious than direct determination of k-functions.Hence, the ksat − ψaev procedure may be a valuable and cost-effective solution in many situations.AcknowledgmentsFunding for this study was provided in part by IndustriesCascades, Inc., Perkins Papers, Ltd., and NSERC (Canada) underthe University-Industry partnership grant number CRD 192179 andthe first author Discovery Grant (NSERC). Support received by thesecond author from the CNPq (Brazil) and from the National Sci-ence Foundation by the last author is also greatly appreciated.ReferencesBurnotte, F. G., Lefebvre, A., Cabral, C. A., and Veilleux, A.,2000, “Use of Deinking Residues for the Final Cover of a MSWLandfill,” Vol. 53e, Conférence Canadienne de Géotechnique,Montreal, pp. 585–591.Cabral, A., Burnotte, F., Lefebvre, G., and Panarotto, C. T., 2002,“Geotechnical Characterization of a Pulp and Paper Residue to beUsed as Alternative Cover Material to Landfill and to Acid Gen-erating Tailings,” Proceedings of the 4th International Congresson Environmental Geotechnics, Rio de Janeiro, Brazil, pp. 207–212.Cabral, A. R., Planchet, L., Marinho, F. A., and Lefebvre, G., 2004,“Determination of the Soil Water Characteristic Curve of HighlyCompressible Materials: Case Study of Pulp and Paper By-Product,” Geotechnical Testing Journal, Vol. 27, pp. 154–162.Fredlund, D. G., 2002, “Use of Soil-Water Characteristic Curve inthe Implementation of Unsaturated Soil Mechanics,” Proceed-ings of the 3rd International Conference on Unsaturated Soils,Recife, Brazil.Fredlund, D. G. and Xing, A. Q., 1994, “Equations for the Soil-Water Characteristic Curve,” Canadian Geotechnical Journal,Vol. 31, pp. 521–532.Huang, S. Y., Barbour, S. L., and Fredlund, D. G., 1998, “Devel-opment and Verification of a Coefficient of Permeability Func-tion for a Deformable Unsaturated Soil,” Canadian GeotechnicalJournal, Vol. 35, pp. 411–425.Laliberte, G. E., Corey, A. T., and Brooks, R. H., 1966, “Prop-erties of Unsaturated Porous Media,” Hydrology Paper No. 17,Colorado State University, Fort Collins, CO.Mualem, Y., 1976, “A New Model for Predicting the HydraulicConductivity of Unsaturated Porous Media,” Water ResourcesResearch, Vol. 12, pp. 513–522.Toll, D. G., 1988, “The Behaviour of Unsaturated Compacted Nat-urally Occurring Gravel,” Ph.D. Thesis, University of London,London.

Determination of the hydraulic conductivity function of a highly compressible material based on tests with saturated samples

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Determination of the hydraulic conductivity function of a highly compressible material based on tests with saturated samples

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