1、 American Society of Civil EngineersStandard Guideline for Fitting SaturatedHydraulic ConductivityUsing Probability Density FunctionsASCE/EWRI 50-08Standard Guideline forCalculating the EffectiveSaturated HydraulicConductivityASCE/EWRI 51-08This document uses both the International System of Units (
2、SI) and customary units.Fitting of Hydraulic Conductivity Using Statistical Spatial EstimationStandards Committee of theStandards Development Council of theEnvironmental and Water Resources Institute of theAmerican Society of Civil Engineers A S C E S T A N D A R DPublished by the American Society o
3、f Civil EngineersASCE/EWRI 50-08ASCE/EWRI 51-08Standard guideline for fitting saturated hydraulic conductivity using probability densityfunctions ASCE/EWRI 50-08 : standard guideline for calculating the effective saturatedhydraulic conductivity ASCE/EWRI 51-08.p. cm.“Fitting of Hydraulic Conductivit
4、y Using Statistical Spatial Estimation StandardsCommittee of the Standards Development Council of the Environmental and WaterResources Institute of the American Society of Civil Engineers.”Includes bibliographical references and index.ISBN 978-0-7844-0993-01. Groundwater flow. 2. Soil permeability.
5、3. Soil moisture. I. American Society ofCivil Engineers. TC176.S73 2008624.15136dc222008037170Published by American Society of Civil Engineers1801 Alexander Bell DriveReston, Virginia 20191www.pubs.asce.orgThis standard was developed by a consensus standards development process which has beenaccredi
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13、und at http:/pubs.asce.org/support/reprints/.Copyright 2008 by the American Society of Civil Engineers.All Rights Reserved.ISBN 13: 978-0-7844-0993-0Manufactured in the United States of America.16 15 14 13 12 11 10 09 08 1 2 3 4 5Library of Congress Cataloging-in-Publication DataSTANDARDSIn 2003, th
14、e Board of Direction approved the revision tothe ASCE Rules for Standards Committees to govern thewriting and maintenance of standards developed by theSociety. All such standards are developed by a consensusstandards process managed by the Societys Codes andStandards Committee (CSC). The consensus p
15、rocessincludes balloting by a balanced standards committeemade up of Society members and nonmembers, ballotingby the membership of the Society as a whole, and ballot-ing by the public. All standards are updated or reaffirmedby the same process at intervals not exceeding five years.The following stan
16、dards have been issued:ANSI/ASCE 1-82 N-725 Guideline for Design andAnalysis of Nuclear Safety Related Earth StructuresASCE/EWRI 2-06 Measurement of Oxygen Transfer inClean WaterANSI/ASCE 3-91 Standard for the Structural Design ofComposite Slabs and ANSI/ASCE 9-91 StandardPractice for the Constructi
17、on and Inspection ofComposite SlabsASCE 4-98 Seismic Analysis of Safety-Related NuclearStructuresBuilding Code Requirements for Masonry Structures(ACI 530-02/ASCE 5-02/TMS 402-02) andSpecifications for Masonry Structures (ACI 530.1-02/ASCE 6-02/TMS 602-02)ASCE/SEI 7-05 Minimum Design Loads for Build
18、ingsand Other StructuresSEI/ASCE 8-02 Standard Specification for the Design ofCold-Formed Stainless Steel Structural MembersANSI/ASCE 9-91 listed with ASCE 3-91ASCE 10-97 Design of Latticed Steel TransmissionStructuresSEI/ASCE 11-99 Guideline for Structural ConditionAssessment of Existing BuildingsA
19、SCE/EWRI 12-05 Guideline for the Design of UrbanSubsurface DrainageASCE/EWRI 13-05 Standard Guidelines for Installationof Urban Subsurface DrainageASCE/EWRI 14-05 Standard Guidelines for Operationand Maintenance of Urban Subsurface DrainageASCE 15-98 Standard Practice for Direct Design ofBuried Prec
20、ast Concrete Pipe Using StandardInstallations (SIDD)ASCE 16-95 Standard for Load Resistance Factor Design(LRFD) of Engineered Wood ConstructionASCE 17-96 Air-Supported StructuresASCE 18-96 Standard Guidelines for In-Process OxygenTransfer TestingASCE 19-96 Structural Applications of Steel Cables for
21、BuildingsASCE 20-96 Standard Guidelines for the Design andInstallation of Pile FoundationsANSI/ASCE/T x0.25and x0.75denote thelower and upper quartiles of K, etc. Calculate the pthquantile using the following equation:(8-3)xpH33527 expH20849Y H11001 zpH9268YH20850H11349fKH20849xH20850 H335271x H1108
22、0 H9268Y H11080 H208572H9266H11080 expH20875H1100212H20873ln x H11002 YH9268YH208742H20876x H11350 0H9268YYH11349H11349H11349H11349H11349H11349in which zpis the pth quantile of a standardized nor-mal variable (Z), that is, with 0 mean and unit standarddeviation(8-4)Commentary. The quantile zpis tabu
23、lated in thetechnical literature. It can be calculated using thespreadsheet Excel, whose function NORMSINV(p)returns the value zpwhen invoked with a numericalvalue for p,0 H11021 p H11021 1. The software Matlab returns zpusing the command norminv(p,0,1).8.3 CALCULATION EXAMPLEAssume that the average
24、 and standard deviation of logconductivity are and and thatthe log conductivity Y is symmetric (in other words,it has skew ). Calculate the quantiles x0.25, x0.5,and x0.75. First, note that the standard-normal quantiles z0.25, z0.5, and z0.75equal H110020.6745, 0, and0.6745, respectively. Second, ap
25、ply Eq. 8-3 to obtainthe desired quantiles x0.25H33527 0.935, x0.5H33527 1.53,x0.75H33527 2.51 in units of length over time.9.0. FITTING K DATA WITH THE (3-PARAMETER) LOG-GAMMA PDF:|CsY| H11022 0.059.1. CALCULATING THE LOG-GAMMA PDFUse the following equation to calculate the log-gammapdf (hK(x):(9-1
26、)where x represents the value of the saturated hydraulicconductivity at which the log-gamma pdf is calculated;and are the shape, scale, and upper or lowerbound parameters of the log-gamma pdf, respectively.The log-gamma pdf may have either a lower bound:if (9-2)or it may have a lower bound (equal to
27、 0) and anupper bound simultaneously, as follows:if (9-3)H9252YH11021 00 H11021 x H11349 eH9258YH9252YH11022 0x H11350 eH9258YH9258YH9252Y,H9251Y,hkH20849xH20850 H33527H20873lnH20849xH20850 H11002H9258YH9252YH20874H9251YH20841lnH20849xH20850 H11002H9258YH20841H110021eH11002H20873lnH20849xH20850H1100
28、2H9258YH9252YH20874xH9003H20849H9251YH20850CsYH33527 0H9268YH33527 0.731,Y H33527 0.426PH20849Z H11349 zpH20850 H33527 pASCE/EWRI 50-085denotes the well-known gamma function,which is defined as follows (v is the variable of inte-gration over the range of positive real numbers):(9-4)The gamma functio
29、n can be calculated using commer-cially available spreadsheets and numerical software.9.2 CALCULATING THE PARAMETERS OF THE LOG-GAMMA PDFLetting , , and CsYbe the sample average, standarddeviation, and coefficient of skew of Y, respectively,calculate estimates of the log-gamma parameters asfollows:(
30、9-5)(9-6)(9-7)9.3 CALCULATING THE QUANTILES OF THELOG-GAMMA DISTRIBUTED HYDRAULICCONDUCTIVITYEquation 9-8 calculates the quantiles (xp) of saturatedhydraulic conductivity as(9-8)in which is defined by the following integral equa-tions (with 0 p 1):if (9-9)in which was defined in Eq. 9-4, and ,orif (
31、9-10)and H9251YH335274CsY2CsYH11021 0H20885H9023q0eH11002vvH9251YH110021dv H33527 1 H11002 p1H9003H20849H9251YH20850H9251YH335274CsY2H9003H20849H9251YH20850CsYH11022 0H20885H9023q0eH11002vvH9251YH110021dv H33527 p1H9003H20849H9251YH20850H11021H11021H9023qxpH33527 expH20875Y H11001H20875H9023qCsY2H11
32、0022CsYH20876H9268YH20876H9258YH33527 Y H110022H9268YCsYH9252YH33527H9268Y CsY2H9251YH335274CsY2H9268YYH9003H20849H9251YH20850 H33527H20885H110090eH11002v vH9251YH110021dvH9003H20849H9251YH20850FITTING SATURATED HYDRAULIC CONDUCTIVITYCommentary. The integration limit in Eqs. 9-9 and 9-10 is the quan
33、tile corresponding to aprobability q of the gamma pdf with shape parameterand in which the scale parameter . The prob-ability qp(if CsY0) or q 1 p (if CsY0).can be calculated using the command gaminv(probability q, , 1) in the software Matlab, or usingthe GAMMAINV (probability q, , 1) function in th
34、esoftware Excel. Rounding off of numbers and thenumber of significant figures used in calculations mayproduce slightly different numerical results, dependingon the choices made by the analyst. When the absolutevalue of the coefficient of skew is less than 0.01, thelikelihood of numerical errors prod
35、uced by commer-cial software should not be overlooked. In thisinstance, it is recommended that calculations be triedin alternative ways to ensure the accuracy of results.9.4 CALCULATION EXAMPLEThe log conductivity data plotted on Fig. 4-2 wereused in this example. The sample average, standarddeviati
36、on, and coefficient of skew of log conductivityare , and , respec-tively. These were used to calculate the parameters, using the equa-tions given in Section 9.2. Because CsYis positive(and, thus, is also positive, see Eq. 9-6), the log-gamma pdf has a lower bound equal to exp( )(see Eq. 9-2). Set ,
37、0.50,0.75 to calculate the quantiles x0.25, x0.50, and x0.75. The values for , 0.50, and 0.75 equal8.98, 11.1, and 13.5, respectively. Use Eq. 9-8 to calculate the values of the desired quantiles:and.10.0 GOODNESS-OF-FIT TESTING10.1 THE CHI-SQUARED TESTThe chi-squared goodness-of-fit test for satura
38、tedhydraulic conductivity is implemented as follows:10.1.1 Step 1Calculate R saturated hydraulic conductivity quan-tiles, denoted by xH9004pH11021 x2H9004pH11021H11021xRH9004p, using Eq. 8-3for lognormal pdf or Eq. 9-8 for log-gamma pdf. Notethat in the notation xrH9004p, the probability correspondi
39、ngto the quantile is rH9004p, in which r H33527 1, 2 R,and,.,.cmH20862sx0.75H33527 5.45 H11003 10H110029,x0.50H33527 1.25 H11003 10H110029,x0.25H33527 3.43 H11003 10H1100210,p H33527 0.25H9023qH33527pq H33527 p H33527 0.251.39 H11003 10H1100212H33527H9258YH9252YH9258YH33527 H1100227.3H9252YH33527 0.
40、616H9251YH33527 11.4,CsYH33527 0.592H9268YH33527 2.08,Y H33527 H1100220.3H9251YH9251YH9023qH11021H11002H33527H11022H33527H9252YH33527 1H9251YH9023q6the probability increment H9004p is defined by Eq. 10-1. A suitable range for R is 4 R 9. The quantilesxrH9004p, r H33527 1, 2 R, are chosen so that the
41、y define R H11001 1 equal-probability, nonoverlapping, intervals ofhydraulic conductivity:P(xrH9004pKx(rH110011)H9004p) H33527 P(K H11021 xH9004p) H33527 P(K H11022 xRH9004p) H33527 H9004p (10-1)for r H33527 1, 2, , R H11002 1, in which(10-2)is the probability of each of the R H11001 1 intervals of
42、saturated hydraulic conductivity defined by the quan-tiles xrH9004p, r H33527 1, 2 R. The quantiles satisfy theprobability statementP(KxrH9004p) H33527 r H33527 1, 2 R (10-3)10.1.2 Step 2The expected number of K measurements that fall in any of the R H11001 1 (equal-probability) intervalsequals , in
43、 which n is the number of K measure-ments available. This number compares with the actualnumber of K measurements observed in the rth inter-val, nr, r H33527 1, 2 R H11001 1. Calculate the test statistic:(10-4)10.1.3 Step 3Determine the chi-squared critical value associ-ated with a 5% significance l
44、evel and degrees offreedom, . The number of degrees of freedomH92732 0.05, RH11002fR H11002 fD H335271n H11080 H9004pH20888RH110011rH335271H20849nrH11002 n H11080 H9004pH208502,.,n H11080 H9004p,.,r H11080 H9004pH11349,.,H9004p H335271R H11001 1H11349H11349,.,H11349H11349of the chi-squared critical
45、value is customarily R.However, parameters ( , ) must be estimatedfrom K data for the lognormal pdf, and parame-ters ( , , ) must be estimated from data for thelog-gamma pdf. Therefore, the number of degrees offreedom of the chi-squared critical value becomes(Benjamin and Cornell 1970, pp. 462466).C
46、ommentary. The chi-squared critical value is tab-ulated in the technical literature. It can also be obtainedusing commercial software. In Excel, the function CHIINV(0.05, ) returns the critical value .The software Matlab returns the critical value using the command chi2inv (0.95, R H11002 f ).10.1.4
47、 Step 4If the test statistic D exceeds , reject thefitted pdf as a suitable probability model for the Kdata. Otherwise, accept the fitted pdf as a suitableprobability model for the K data.10.2 CALCULATION EXAMPLEThe goodness-of-fit of the log-gamma pdf to the Kdata shown in Fig. 4-1 is assessed. The
48、 statistics per-taining to the K data are found in Section 9.4. ChooseR H33527 9. Quantify nine quantiles xH9004pH11021 x2H9004pH11021 x9H9004p,using Eq. 9-8, which defines R H11001 1 H33527 9 H11001 1 H33527 10equal-probability intervals. The probability associatedwith each interval is H9004p H3352
49、7 1H2086210 H33527 0.10, so that theexpected number of K measurements in each intervalis The number of meas-urements observed in each interval is counted from theK sample. The test statistic (D in Eq. 10-4) is calcu-lated, and the chi-squared critical value is determined.Results are summarized in Table 10-1.n H11080 H9004p H33527 201 H11003 0.10 H33527 20.1.H92732 0.05, RH11002fH92732 0.05, RH11002fH92732 0.05, RH11002fR H11002 fR H11002 fH9258YH9252YH9251Yf H33527 3H9268YYf H33527 2ASCE/EWRI 50-08
