Research ArticleSensitivity Analysis and Validation forNumerical Simulation of Water Infiltration intoUnsaturated Soil
Eng Giap Goh12 and Kosuke Noborio3
1Graduate School of Agriculture Meiji University 1-1-1 Higashimita Tama-ku Kawasaki 214-8571 Japan2School of Ocean Engineering Universiti Malaysia Terengganu 21030 Kuala Terengganu Terengganu Malaysia3School of Agriculture Meiji University 1-1-1 Higashimita Tama-ku Kawasaki 214-8571 Japan
Correspondence should be addressed to Kosuke Noborio noboriokiscmeijiacjp
Received 23 July 2015 Accepted 31 August 2015
Academic Editor Robert Cerny
Copyright copy 2015 E G Goh and K Noborio This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A FORTRAN code for liquid water flow in unsaturated soil under the isothermal condition was developed to simulate waterinfiltration into Yolo light clay The governing equation that is Richardsrsquo equation was approximated by the finite-differencemethod A normalized sensitivity coefficient was used in the sensitivity analysis of Richardsrsquo equation Normalized sensitivitycoefficient was calculated using one-at-a-time (OAT) method and elementary effects (EE) method based on hydraulic functionsfor matric suction and hydraulic conductivity Results from EE method provided additional insight into model input parameterssuch as input parameter linearity and oscillating sign effect Boundary volumetric water content (120579
119871(upper bound)) and saturated
volumetric water content (120579119904) were consistently found to be the most sensitive parameters corresponding to positive and negative
relations as given by the hydraulic functions In addition although initial volumetric water content (120579119871(initial cond)) and time-
step size (Δt) respectively possessed a great amount of sensitivity coefficient and uncertainty value they did not exhibit significantinfluence on model output as demonstrated by spatial discretization size (Δz) The input multiplication of parameters sensitivitycoefficient and uncertainty value was found to affect the outcome of model simulation in which parameter with the highest valuewas found to be Δz
1 Introduction
Sensitivity analysis is used for various reasons such asdecision-making or development of recommendations com-munication increasing understanding or quantification ofsystem and model development In model development itcan be used for the purposes of model validation or accuracysimplification calibration and coping with poor or missingdata and even to identify important parameter for furtherstudies [1]
More than a dozen sensitivity analysis methods areavailable ranging from one-at-a-time (OAT) to variance-based methods [2 3] In a fundamental level sensitivityanalysis is a tool to assess the effect of changes in inputparameter value on output value of a simulation model In
this aspect the sensitivity coefficient in a normalized formis given in the following relation
119883119894119908
=120597119910119894119910119894
120597119886119908119886119908
(1)
where 119883119894119908
is referred to as normalized sensitivity coefficientfor 119908th input parameter at 119894th observation point 119910
119894is model
dependent variable value at 119894th observation point and 119886119908is
119908th input parameter valueThis method utilizes derivative ata single point and similarly it can be applied as OAT methodwhen one input parameter is varied while holding otherparameters fixed However the former and the lattermethodsdo not explore other input space factors in which more thanone input parameter is varied Despite this disadvantage
Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2015 Article ID 824721 7 pageshttpdxdoiorg1011552015824721
2 International Scholarly Research Notices
Saltelli and Annoni [4] noticed that researchers continuouslypractice OATmethod mainly due to few advantages claimedfor example a safe starting point where the model propertiesare well known and all OAT sensitivities relative to a startingpoint Although variance-based method is the best practiceSaltelli and Annoni [4] have suggested the use of elementaryeffects method which is an enhancement of OAT methodwhen computation time is expensive for instance in numer-ical simulation that is computationally demanding
Elementary effects method is accomplished through theuse of a technical scheme to generate trajectories Eachtrajectory consists of a number of steps in which each step isreferred to an increment or decrement of an input parametervalueThe base condition for each trajectory is different fromthe others and it is selected randomly The random versionof trajectory generation is as follows [5]
Blowast = J119896+1119896
xlowast + (Δ
2) [(2B minus J
119896+1119896)Dlowast + J
119896+1119896]Plowast (2)
where Blowast is generated trajectory in the form of matrix withdimension (119896 + 1) times 119896 where 119896 is the number of independentinput parameters Δ is a value in [1(119901 minus 1) 1 minus 1(119901 minus 1)]
and 119901 is the number of levels J119896+1119896
is (119896 + 1) times 119896 matrix of1rsquos xlowast is a randomly chosen base value B is lower triangularmatrix of 1rsquos Dlowast is 119896-dimensional diagonal matrix in whicheach element is either +1 or minus1 by random generation andPlowast is 119896-by-119896 random permutation matrix that each row andcolumn of the matrix with only one element equal to 1 whilethe other elements of the matrix are zero
The generated trajectories can be screened to obtain asubset of trajectories with the greatest geometric distancesThe trajectories scanning to maximize geometric distancesbetween all the pairs of points between two trajectories is asfollows [6]
119889119898119897
=
119896+1
sum
119894=1
119896+1
sum
119895=1
radic
119896
sum
119911=1
[119883(119894)
119911=1(119898) minus 119883
(119895)
119911=1(119897)]2
for 119898 = 119897
0 for 119898 = 119897
(3)
where 119889119898119897
is distance between a pair of trajectories 119898 and 119897119883(119895)
119911=1(119897) is 119911th coordinate of the 119895th point of the 119897th trajectory
and 119883(119894)
119911=1(119898) is 119911th coordinate of the 119894th point of the 119898th
trajectoryThe sensitivity coefficient of an input parameter in ele-
mentary effects method is presented as 120583119894 which is the mean
of elementary effects (119864119864119895119894) 120583119894
lowast is the mean of absolute valuesof the elementary effects which is used to avoid cancellationof difference signs in themean valueThe sensitivitymeasures(120583119894 120583119894
lowast and 120590) and 119864119864119895
119894are given by [5]
119864119864119895
119894=119910119895(119909119894+ Δ119894) minus 119910119895(119909119894)
where 119910119895(119909119894) and 119910119895(119909
119894+Δ119894) are simulation result before and
after increment or decrement ofΔ value that isΔ119894 which can
be either of positive or negative value 119903 is the total number oftrajectories 119864119864119895
119894is elementary effects of 119894 input parameter at
119895 trajectory and 120590119894is standard deviation of 119894 input parameter
The aim of the current work is to carry out sensitivityanalysis on water infiltration into unsaturated soil as gov-erned by Richardsrsquo equation that is governing equation ofsoil water flow and use it as an evaluating method to validatethe simulation source code with analytical solutionThus theobjectives of this study are to (1) determine the sensitivitycoefficient and (2) to validate model simulation with Philiprsquossemianalytical solution from literatures using the sensitivitycoefficient under a hypothetical assumption In this study weused the water infiltration results from Haverkamp et al [7]and Kabala and Milly [8] to verify the simulation
2 Materials and Methods
21 The Governing Equation of Water Flow in UnsaturatedSoil and Its Numerical Solution The governing equation fortransient liquid water flow in soil may be described as [9]
120597120579119871
120597119905=
120597
120597119911[(119870
120597120595119898
120597120579119871
)120597120579119871
120597119911minus 119870
997888119896] (8)
where 120579119871is volumetric water content (m3mminus3) 119905 is time (s) 119911
with a value of positive one when it is vertically downwardsEquation (8) was approximated numerically and its alge-
bra was implemented in FORTRAN 2008 using Simply FOR-TRAN Integrated Development Environment The spatialdiscretization method used is termed as cell-centered finite-difference and the temporal discretization method used wasthe fully implicit scheme In order to avoid unnecessaryredundancy we only provide the algebra for (8) that is usedfor sensitivity analysis in the current study as follows
where 119896 indicates a cell-centered number in 119911-directionin Cartesian coordinate system Δ119905 (s) is time-step size120579119871(119896)
119899 (m3mminus3) and 120579119871(119896)
119899+1 (m3mminus3) are volumetric watercontent at old time level (119899) and new time level (119899 + 1)respectively 119870
119896+12(m sminus1) is hydraulic conductivity at the
interface between cells 119896 and 119896+1119870119896minus12
(m sminus1) is hydraulicconductivity at the interface between cells 119896 minus 1 and 119896(120597120595119898120597120579119871)119896+12
is partial derivative of 120595119898with respect to 120579
119871
at the interface between cells 119896 and 119896 + 1 (120597120595119898120597120579119871)119896minus12
ispartial derivative of 120595
119898with respect to 120579
119871at the interface
between cells 119896 minus 1 and 119896 Δ119911119896+1
(m) Δ119911119896(m) and Δ119911
119896minus1(m)
are corresponding to the spatial sizes of spacing of cells 119896+1 119896and 119896minus1 respectively 120579
119871(119896+1)
119899+1 (m3mminus3) 120579119871(119896)
119899+1 (m3mminus3)and 120579
119871(119896minus1)
119899+1 (m3mminus3) are the volumetric water contents atnew time level of cells 119896+1 119896 and 119896minus1 respectively Equation(8) was numerically solved by a fully implicit cell-centeredfinite-difference scheme without any linearization An itera-tive method was used to solve the mathematical algebra of(9) that is Jacobi iteration [10] For comparison purposemodified Newton-Raphson method was also implemented[11] A convergence factor criterion was used to indicate thecondition for iteration termination process that is absolutemaximum difference |120579
119871(119896)
119899+1minus 120579119871(119896)
119899| for every single cell
22 The Constitutive Functions of Matric Pressure Head (120595119898)
and Hydraulic Conductivity (119870) The hydraulic functionsused were adopted from Haverkamp et al [7]
where 120572 120573 119860 and 119861 are fitting parameters 120579119903(m3mminus3)
is residual volumetric water content 120579119904(m3mminus3) is satu-
rated volumetric water content and 119870119904(m sminus1) is saturated
hydraulic conductivity
23 Numerical Experiment and the Default Setting of InputParameters of the Flow Problem Water infiltration intoYolo light clay was used in the numerical experiment Thehydraulic functions for the soil (see (10)) and the coefficientsvalues are shown in Table 1 Initial condition for the volumet-ric water content was 02376m3mminus3 Lower boundary wasset as free-drainage to water flow Upper boundary was setat 0495m3mminus3 After considering the mass balance ratio [9]and iteration number the time-step size spatial discretizationsize and convergent value were set at corresponding valuesof 500 s 1 cm and 10minus12m3mminus3 respectively The iterationmethods of Jacobi and modified Newton-Raphson werecompared It was found that the minimum iteration numberfrom the latter was equivalent to the iteration number fromthe former when the relaxation factor of the latter was setto unity (data not shown) Reducing the relaxation factorfrom unity would result in increasing iteration number Thenumerical solution of (9) did not exhibit convergent problemthus Jacobi iteration method was sufficient
Table 1 The coefficient values from Haverkamp et al (1977) [7]based on (10) These values were used as base case Note that 120579
119903is
residual volumetric water content 120579119904is saturated volumetric water
content 119870119904is saturated hydraulic conductivity and 120572 120573 119860 and 119861
are fitting coefficients
Parameter Value120572 739120579119903
0124m3mminus3
120579119904
0495m3mminus3
120573 4119860 1246119861 177119870119904
123 times 10minus7msminus1
00
50
100
150
200
250
300
020 025 030 035 040 045 050
Simulated 105 s
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Philip(H) at 105 sPhilip(K) at 105 s
Figure 1 Comparison of simulated results with Philiprsquos semianalyt-ical solution Philip(H) and Philip(K) were from Haverkamp et al[7] and Kabala and Milly [8] respectively
24 Statistical Measures In order to determine the goodnessof fit between reference data and simulated results onestatistical equation was implemented The equation is calledabsolute residual errors (MA) as follows [12]
MA =1
119873
119873
sum
119896=1
1003816100381610038161003816cal119896 minus obs119896
1003816100381610038161003816 (11)
where cal119896is the simulated data at cell 119896 and obs
119896is the
analytical solution as reference data at cell 119896
3 Results and Discussion
31 Simulation Results and Their Accuracy Based on theconditions as stated in previous section water infiltration intoYolo light clay was simulated up to 105 s Data on Philiprsquossemianalytical solution were collected fromHaverkamp et al[7] hereafter referred to as Philip(H) Simulation results werecompared with the data to verify the simulation (Figure 1)It was evident that the simulation results slightly underpre-dicted the infiltration front of water flow
To further reinforce the previous claim some datawere extracted from Kabala and Milly [8] as indicated by
4 International Scholarly Research Notices
Philip(K) as in Figure 1 for further comparison Figure 1shows that there was a small difference between Philip(K)and Philip(H) but the former was relatively closer to thesimulation results than the latter At this point of observationwe were unable to determine which of the solutions thatis Philip(K) and Philip(H) provided from the literature wasaccurateHowever results from the figure clearly indicate thatthe simulated result was lesser than Philiprsquos semianalyticalsolution Therefore sensitivity analysis was carried out todetermine the sensitivity coefficient for all input parametersand use the sensitivity analysis results to assess the modelsimulation based on the assumption that possibly the cumu-lative effect of input parameters in terms of significant digitsapproximation could be contributing to the underpredictionof the volumetric water content of the simulation In additionsensitivity analysis is one of the most important steps inevaluating the effect of input parameter on simulation resultsand it is also used by others for model validation [13ndash16]
32 Sensitivity Analysis and Simulation Model ValidationNegligible sensitivity response could be due to too small per-turbation size and inaccuracy in sensitivity response couldbe due to too large perturbation size [17] Values of inputparameters were subjected to a perturbation size betweenminus5 and 5 as suggested by Zheng and Bennett [12] andin considering the simulation time we limit the sensitivityanalysis to a simulation time of 105 s The sensitivity analysisstudy was based on a single perturbation size of incrementor decrement in each simulation The sensitivity analysiswas carried out based on the hydraulic functions (10) fromHaverkamp et al [7]
There were seven input parameters from Haverkamphydraulic functions as listed in Table 1 Additional fourinput parameters were also tested that is initial volumetricwater content (120579
119871(initial cond)) boundary volumetric water
content 120579119871(upper cond) time-step size (Δ119905) and spatial
spacing size (Δ119911) The depth at 155 cm from the groundsurface was used for observation
The normalized sensitivity coefficients are shown inFigure 2 Generally there are two groups of sensitivity coef-ficients that is positive and negative relations In positiverelation group the boundary volumetric water content hadthe highest sensitivity coefficientThis was followed by initialvolumetric water content and saturated hydraulic conductiv-ity The smallest sensitivity coefficient in the group was theresidual volumetric water content In negative relation groupsaturated volumetric water content had the highest sensitivitycoefficient and this group endedwith spatial spacing size andtime-step size as the smallest sensitivity coefficient
For comparison purpose elementary effects method wasalso used to calculate normalized sensitivity coefficient Weassumed only random generation in 119896-dimensional diagonalmatrix (Dlowast) and then (2) was used to generate 50 trajectoriesEquation (3) was used to screen out 4 trajectories with thegreatest geometric distance of those trajectories Equations(4) to (7) were used to calculate the elementary effects meanof elementary effects mean of absolute values of the elemen-tary effects and standard deviation respectivelyThemean ofelementary effects was modified to calculate the normalized
100300500700900
1100
Input parameter
minus100
minus300
minus500
minus700
minus900
minus1100
Nor
mal
ized
sens
itivi
ty co
effici
ent (
)
120579L
(upp
er b
ound
)
120579L
(initi
al co
nd)120579s 120579r
Ks120573 A120572Δz
ΔtB
644Eminus01
508E
minus01
208Eminus01
825Eminus02
385Eminus02
minus619Eminus04
minus550E
minus02
minus589Eminus01
minus107E+00
minus497E+00
471E
+00
Figure 2 The rank of sensitivity coefficient Note 120579119904and 120579
119903
are saturated and residual volumetric water content Δ119911 spatialspacing sizeΔ119905 time-step size119870
119904 saturated hydraulic conductivity
120579119871(initial cond) clay medium initial value of volumetric water
content 120579119871(upper bound) upper boundary of volumetric water
content 119860 119861 120573 and 120572 are fitting parameters from Haverkamp asin (10)
Table 2 Statistical measures (120583 120583lowast and 120590) of elementary effectsmethod They are the mean of elementary effects the meanof absolute values of the elementary effects and the standarddeviation respectively Note that 120579
119903is residual volumetric water
content 120579119904is saturated volumetric water content 119870
119904is saturated
hydraulic conductivityΔ119911 is spatial spacing sizeΔ119905 is time-step size120579119871(initial cond) is initial value of volumetric water content and 120572
120573 119860 and 119861 are fitting coefficients
120583 () 120583lowast () 120590
120579119904
minus603119864 + 00 603119864 + 00 948119864 minus 01
119861 minus185119864 + 00 185119864 + 00 938119864 minus 01
120573 minus207119864 minus 01 320119864 minus 01 370119864 minus 01
120572 minus414119864 minus 02 125119864 minus 01 147119864 minus 01
Δ119911 minus335119864 minus 02 395119864 minus 02 456119864 minus 02
Δ119905 minus166119864 minus 04 525119864 minus 04 621119864 minus 04
120579119903
444119864 minus 03 337119864 minus 02 402119864 minus 02
119860 313119864 minus 01 313119864 minus 01 741119864 minus 02
119870119904
524119864 minus 01 524119864 minus 01 673119864 minus 02
120579119871(initial cond) 884119864 minus 01 884119864 minus 01 305119864 minus 01
sensitivity coefficient The results are shown in Table 2 Thesensitivity coefficient has identical ranking as those obtainedin Figure 2 except for the coefficient of 120572 input parameterSimilar values of 120583 and 120583
lowast indicate linear effect on few inputparameters in positive (119860119870
119904 and 120579
119871(initial cond)) and nega-
tive (120579119904and 119861) relations Other input parameters have shown
the effect of oscillating sign that results in different values of120583 and 120583
lowast In general those sensitivity coefficients generatedby different methods have shown comparable results
We assumed that a minor deviation in each input param-eter in terms of its significant digits approximation couldcontribute some effects on the simulation outcome thatcould possibly explain the discrepancy between the simulatedresults and Philiprsquos semianalytical solution (Figure 1) In other
International Scholarly Research Notices 5
Table 3 Significant digits approximation on input parameter valueNote that 120579
119903is residual volumetric water content 120579
119904is saturated
volumetric water content 119870119904is saturated hydraulic conductivity
Δ119911 is spatial spacing size Δ119905 is time-step size 120579119871(initial cond) is
initial value of volumetric water content 120579119871(upper bond) is upper
boundary of volumetric water content and 120572 120573119860 and 119861 are fittingcoefficients
Δ119905 10 s the base case was 500 sΔ119911 01 cm the base case was 1 cm
words the parameter values in terms of significant digitsapproximation that were used in computer simulation byHaverkamp et al [7] could be different from the exactdata in terms of input parameter significant digits thatthey published Thus we take advantage on the positive andnegative relations generated from the sensitivity analysis andset up a hypothetical approximation value in Table 3 forfurther investigation The cumulative effect was studied bymanipulating an input parameter used for each simulationand the subsequent manipulation of input parameter wascarried out on top of the previous changed input parameterThis process begins from step 1 for base case to step 10 forspatial spacing size For instance the 120579
119871(initial cond) value
(02376499m3mminus3) was used as a second simulation (in step2) after the base case simulation This was followed by thirdsimulation (in step 3) using 120579
119903value as 0124499m3mminus3
by remaining 120579119871(initial cond) value used in the second
simulation For each simulation Equation (11) was used tocalculate the discrepancy between simulation results andPhiliprsquos semianalytical solution (data from [7]) for absoluteresidual error (MA)Of all those eleven parameters inTable 3Δ119905 and Δ119911 were the only two parameters without any limit ofvariation and for this reason we extend the variation limitby reducing the former and the latter by 98 and 90 from500 s and 1 cm to 10 s and 01 cm respectively The 120579
119904and
120579119871(119906119901119901119890119903 119887119900119906119899119889) values are negative and positive relations
respectively Decreasing and increasing the correspondingformer and latter values would result in simulation failurethus those two parameters remained unchanged
A consistent reduction inMA value from 120579119871(initial cond)
to 119861 input parameter was observed except a slight incrementat Δ119905 input parameter simulation and a steep slide of MAvalue was observed on the Δ119911 input parameter simulation(Figure 3) Although the sensitivity coefficient in Figure 2indicates that reducing Δ119905 value should lead to a reductionin MA value the simulated result showed an increase in
002
54
002
53
002
53
002
45
002
45
002
45
002
45
002
338
002
343
001
06
0010
0015
0020
0025
0030
0035
Input parameter
Abso
lute
resid
ual e
rror
(MA
)
Δz
(ste
p10
)
Δt
(ste
p9
)
B(s
tep8
)
A(s
tep7
)
Ks
(ste
p6
)
120572(s
tep5
)
120573(s
tep4
)
120579L
(initi
al co
nd)
(ste
p2
)
Base
case
(ste
p 1)
120579r
(ste
p3
)
(m3
mminus3)
Figure 3The cumulative effect of input parameters on the absoluteresidual error at simulation time 105 s Note 120579
119871(initial cond) (step
2) claymedium initial value of volumetric water content 120579119903(step 3)
residual volumetric water content 120573 (step 4) 120572 (step 5) 119860 (step 7)and 119861 (step 8) are fitting parameters119870
the MA value This observation could be explained from theresult of elementary effects method This was because Δ119905 hasdifferent values of 120583 and 120583
lowast which indicate the capability ofsign oscillation (Table 2)
Figure 3 shows that the simulation on the cumulativeeffect of steps 2ndash9 which combined the effect from 120579
119871(initial
cond) (step 2) with Δ119905 (step 9) did not contribute to anysignificant effects on the advancement of water infiltrationfront It only resulted in a reduction of 78 in MA valuefrom 00254 to 002343m3mminus3 In addition those eight inputparameters had to vary in significant digits approximation astabulated in Table 3 in order to result in the stated percentagereduction Therefore the significant digits approximationmight not be the main cause of the problem in consideringthat a more significant effect on the advance of water infiltra-tion front was shown by Δ119911 in the Figure 4 A further step toinclude Δ119911 in the simulation that is the cumulative effect ofsteps 2ndash10 which combined the effect from 120579
119871(initial cond)
(step 2) with Δ119911 (step 10) there was 547 reduction in MAvalue of step 9 from 002343 to 00106m3mminus3 This indicatesthat the spatial spacing size was themain cause in the advanceof water infiltration front Therefore the simulation wasrepeated for the last time for the effect of spatial spacing sizealone and in Figure 4 there was a good agreement betweenthe simulation results and the Philip(K) This observationcould be explained using (1) after rearranging it into thefollowing form which we termed as percentage variation insimulation results
Δ119910119894
119910119894
=Δ119886119908
119886119908
119883119894119908 (12)
6 International Scholarly Research Notices
16
17
18
19
20
21
22
23
027 029 031 033 035 037 039
Philip(H)
Base case Philip(K)Only step 10
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Steps 2ndash10Steps 2ndash9
Figure 4 The effect of Δ119911 (step 10 alone) and cumulative effects ofsteps 2 to 9 and 2ndash10 in comparison with Philip(H) and Philip(K)
where Δ119886119908119886119908
is the normalized input parameter valueΔ119910119894119910119894is the normalized output parameter value and 119883
119894119908
is the normalized sensitivity coefficient () Equation (12)is simply a multiplication of the percentage change in inputparameter value from the base case and the normalizedsensitivity coefficient
Using (12) the percentage variation in simulation resultsfrom input parameters of Δ119911 and Δ119905 caused an incrementof 495 and 006 respectively despite Δ119905 having thehighest reduction in percentage (minus98) from base case Thisobservation could be summarized as follows firstly inputparameter with the highest sensitivity coefficient does notguarantee the greatest effect on the simulation result forexample 120579
119871(initial cond) secondly input parameter with
the highest percentage of change also does not guaranteethe greatest effect on the simulation result for exampleΔ119905 and therefore only the highest sensitivity coefficientwith the highest percentage change on input parameter (orthe uncertainty) would give the most substantial effect onsimulation result
4 Conclusions
Thegoverning equation of transientwater flow in unsaturatedand nonisothermal conditions was approximated numeri-cally by finite-difference solution It was successfully imple-mented into FORTRAN programming language simulatedand verified by Philiprsquos semianalytical solution on waterinfiltration into Yolo light clay with data from literatures
One-at-a-timeOAT and elementary effects EEmethodswere used in the sensitivity analysis A common trend ofsensitivity was observed across the methods in both positiveand negative relationsThe latter method allowed explorationof additional characteristics of input parameters at differentinput space such as linearity and sign oscillation effectThe sign oscillation effect observed on input parametersexplained the possibility of its deviation from those observedin OAT method at different input spaces
A hypothetical case that was established to study thecumulative effect of input parameters on the discrepancybetween simulated result and Philiprsquos semianalytical solutionin terms of significant digits approximation (from base case)was found to be unlikely A large normalized sensitivitycoefficient was made with initial volumetric water contentand the largest percentage changes were with time-step sizebut surprisingly none of them contributes to any substantialimpact on simulation results when compared to spatialspacing size This observation led to the conclusion that theuncertainty of input parameter and normalized sensitivitycoefficient of input parameters both controlled the outcomeof simulation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportfrom Meiji University Japan and the Ministry of EducationMalaysia and alsoUniversitiMalaysia TerengganuMalaysia
References
[1] D J Pannell ldquoSensitivity analysis of normative economic mod-els theoretical framework and practical strategiesrdquoAgriculturalEconomics vol 16 no 2 pp 139ndash152 1997
[2] D M Hamby ldquoA review of techniques for parameter sensitivityanalysis of environmental modelsrdquo Environmental Monitoringand Assessment vol 32 no 2 pp 135ndash154 1994
[3] A Saltelli S Tarantola F Campolongo andMRatto SensitivityAnalysis in Practice A Guide to Assessing Scientific Models JohnWiley amp Sons Hoboken NJ USA 2004
[4] A Saltelli and P Annoni ldquoHow to avoid a perfunctory sensitiv-ity analysisrdquo Environmental Modelling and Software vol 25 no12 pp 1508ndash1517 2010
[5] A Saltelli M Ratto and T Andres Global Sensitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008
[6] F Campolongo J Cariboni and A Saltelli ldquoAn effectivescreening design for sensitivity analysis of large modelsrdquo Envi-ronmental Modelling and Software vol 22 no 10 pp 1509ndash15182007
[7] R Haverkamp M Vauclin J Touma P J Wierenga and GVachaud ldquoA comparison of numerical simulation models forone-dimensional infiltrationrdquo Soil Science Society of AmericaJournal vol 41 no 2 pp 285ndash294 1977
[8] Z J Kabala and P C D Milly ldquoSensitivity analysis of flowin unsaturated heterogeneous porous media theory numericalmodel and its verificationrdquo Water Resources Research vol 26no 4 pp 593ndash610 1990
[9] M A Celia E T Bouloutas and R L Zarba ldquoA generalmass-conservative numerical solution for the unsaturated flowequationrdquo Water Resources Research vol 26 no 7 pp 1483ndash1496 1990
[10] J Tu G H Yeoh and C Liu Computational Fluid DynamicsA Practical Approach Butterworth-Heinemann Oxford UK2008
International Scholarly Research Notices 7
[11] J Istok ldquoStep 4 solve system of equationsrdquo in GroundwaterModeling by the Finite Element Method chapter 5 pp 176ndash225American Geophysical Union Washington DC USA 1989
[12] C Zheng and G D Bennett Applied Contaminant TransportModeling JohnWiley amp Sons New York NY USA 2nd edition2002
[13] F Stange K Butterbach-Bahl H Papen S Zechmeister-Boltenstern C Li and J Aber ldquoA process-oriented model ofN2O and NO emissions from forest soils 2 Sensitivity analysis
and validationrdquo Journal of Geophysical Research Atmospheresvol 105 no 4 Article ID 1999JD900948 pp 4385ndash4398 2000
[14] R Nathan U N Safriel and I Noy-Meir ldquoField validation andsensitivity analysis of a mechanistic model for tree seed dis-persal by windrdquo Ecology vol 82 no 2 pp 374ndash388 2001
[15] C HMin Y L He X L Liu B H YinW Jiang andW Q TaoldquoParameter sensitivity examination and discussion of PEM fuelcell simulation model validation Part II Results of sensitivityanalysis and validation of the modelrdquo Journal of Power Sourcesvol 160 no 1 pp 374ndash385 2006
[16] S N Gosling and N W Arnell ldquoSimulating current globalriver runoff with a global hydrological model model revisionsvalidation and sensitivity analysisrdquo Hydrological Processes vol25 no 7 pp 1129ndash1145 2011
[17] E P Poeter and M C Hill Documentation of UCODE A Com-puter Code for Universal Inverse Modeling vol 98 DIANEPublishing 1998
Saltelli and Annoni [4] noticed that researchers continuouslypractice OATmethod mainly due to few advantages claimedfor example a safe starting point where the model propertiesare well known and all OAT sensitivities relative to a startingpoint Although variance-based method is the best practiceSaltelli and Annoni [4] have suggested the use of elementaryeffects method which is an enhancement of OAT methodwhen computation time is expensive for instance in numer-ical simulation that is computationally demanding
Elementary effects method is accomplished through theuse of a technical scheme to generate trajectories Eachtrajectory consists of a number of steps in which each step isreferred to an increment or decrement of an input parametervalueThe base condition for each trajectory is different fromthe others and it is selected randomly The random versionof trajectory generation is as follows [5]
Blowast = J119896+1119896
xlowast + (Δ
2) [(2B minus J
119896+1119896)Dlowast + J
119896+1119896]Plowast (2)
where Blowast is generated trajectory in the form of matrix withdimension (119896 + 1) times 119896 where 119896 is the number of independentinput parameters Δ is a value in [1(119901 minus 1) 1 minus 1(119901 minus 1)]
and 119901 is the number of levels J119896+1119896
is (119896 + 1) times 119896 matrix of1rsquos xlowast is a randomly chosen base value B is lower triangularmatrix of 1rsquos Dlowast is 119896-dimensional diagonal matrix in whicheach element is either +1 or minus1 by random generation andPlowast is 119896-by-119896 random permutation matrix that each row andcolumn of the matrix with only one element equal to 1 whilethe other elements of the matrix are zero
The generated trajectories can be screened to obtain asubset of trajectories with the greatest geometric distancesThe trajectories scanning to maximize geometric distancesbetween all the pairs of points between two trajectories is asfollows [6]
119889119898119897
=
119896+1
sum
119894=1
119896+1
sum
119895=1
radic
119896
sum
119911=1
[119883(119894)
119911=1(119898) minus 119883
(119895)
119911=1(119897)]2
for 119898 = 119897
0 for 119898 = 119897
(3)
where 119889119898119897
is distance between a pair of trajectories 119898 and 119897119883(119895)
119911=1(119897) is 119911th coordinate of the 119895th point of the 119897th trajectory
and 119883(119894)
119911=1(119898) is 119911th coordinate of the 119894th point of the 119898th
trajectoryThe sensitivity coefficient of an input parameter in ele-
mentary effects method is presented as 120583119894 which is the mean
of elementary effects (119864119864119895119894) 120583119894
lowast is the mean of absolute valuesof the elementary effects which is used to avoid cancellationof difference signs in themean valueThe sensitivitymeasures(120583119894 120583119894
lowast and 120590) and 119864119864119895
119894are given by [5]
119864119864119895
119894=119910119895(119909119894+ Δ119894) minus 119910119895(119909119894)
where 119910119895(119909119894) and 119910119895(119909
119894+Δ119894) are simulation result before and
after increment or decrement ofΔ value that isΔ119894 which can
be either of positive or negative value 119903 is the total number oftrajectories 119864119864119895
119894is elementary effects of 119894 input parameter at
119895 trajectory and 120590119894is standard deviation of 119894 input parameter
The aim of the current work is to carry out sensitivityanalysis on water infiltration into unsaturated soil as gov-erned by Richardsrsquo equation that is governing equation ofsoil water flow and use it as an evaluating method to validatethe simulation source code with analytical solutionThus theobjectives of this study are to (1) determine the sensitivitycoefficient and (2) to validate model simulation with Philiprsquossemianalytical solution from literatures using the sensitivitycoefficient under a hypothetical assumption In this study weused the water infiltration results from Haverkamp et al [7]and Kabala and Milly [8] to verify the simulation
2 Materials and Methods
21 The Governing Equation of Water Flow in UnsaturatedSoil and Its Numerical Solution The governing equation fortransient liquid water flow in soil may be described as [9]
120597120579119871
120597119905=
120597
120597119911[(119870
120597120595119898
120597120579119871
)120597120579119871
120597119911minus 119870
997888119896] (8)
where 120579119871is volumetric water content (m3mminus3) 119905 is time (s) 119911
with a value of positive one when it is vertically downwardsEquation (8) was approximated numerically and its alge-
bra was implemented in FORTRAN 2008 using Simply FOR-TRAN Integrated Development Environment The spatialdiscretization method used is termed as cell-centered finite-difference and the temporal discretization method used wasthe fully implicit scheme In order to avoid unnecessaryredundancy we only provide the algebra for (8) that is usedfor sensitivity analysis in the current study as follows
where 119896 indicates a cell-centered number in 119911-directionin Cartesian coordinate system Δ119905 (s) is time-step size120579119871(119896)
119899 (m3mminus3) and 120579119871(119896)
119899+1 (m3mminus3) are volumetric watercontent at old time level (119899) and new time level (119899 + 1)respectively 119870
119896+12(m sminus1) is hydraulic conductivity at the
interface between cells 119896 and 119896+1119870119896minus12
(m sminus1) is hydraulicconductivity at the interface between cells 119896 minus 1 and 119896(120597120595119898120597120579119871)119896+12
is partial derivative of 120595119898with respect to 120579
119871
at the interface between cells 119896 and 119896 + 1 (120597120595119898120597120579119871)119896minus12
ispartial derivative of 120595
119898with respect to 120579
119871at the interface
between cells 119896 minus 1 and 119896 Δ119911119896+1
(m) Δ119911119896(m) and Δ119911
119896minus1(m)
are corresponding to the spatial sizes of spacing of cells 119896+1 119896and 119896minus1 respectively 120579
119871(119896+1)
119899+1 (m3mminus3) 120579119871(119896)
119899+1 (m3mminus3)and 120579
119871(119896minus1)
119899+1 (m3mminus3) are the volumetric water contents atnew time level of cells 119896+1 119896 and 119896minus1 respectively Equation(8) was numerically solved by a fully implicit cell-centeredfinite-difference scheme without any linearization An itera-tive method was used to solve the mathematical algebra of(9) that is Jacobi iteration [10] For comparison purposemodified Newton-Raphson method was also implemented[11] A convergence factor criterion was used to indicate thecondition for iteration termination process that is absolutemaximum difference |120579
119871(119896)
119899+1minus 120579119871(119896)
119899| for every single cell
22 The Constitutive Functions of Matric Pressure Head (120595119898)
and Hydraulic Conductivity (119870) The hydraulic functionsused were adopted from Haverkamp et al [7]
where 120572 120573 119860 and 119861 are fitting parameters 120579119903(m3mminus3)
is residual volumetric water content 120579119904(m3mminus3) is satu-
rated volumetric water content and 119870119904(m sminus1) is saturated
hydraulic conductivity
23 Numerical Experiment and the Default Setting of InputParameters of the Flow Problem Water infiltration intoYolo light clay was used in the numerical experiment Thehydraulic functions for the soil (see (10)) and the coefficientsvalues are shown in Table 1 Initial condition for the volumet-ric water content was 02376m3mminus3 Lower boundary wasset as free-drainage to water flow Upper boundary was setat 0495m3mminus3 After considering the mass balance ratio [9]and iteration number the time-step size spatial discretizationsize and convergent value were set at corresponding valuesof 500 s 1 cm and 10minus12m3mminus3 respectively The iterationmethods of Jacobi and modified Newton-Raphson werecompared It was found that the minimum iteration numberfrom the latter was equivalent to the iteration number fromthe former when the relaxation factor of the latter was setto unity (data not shown) Reducing the relaxation factorfrom unity would result in increasing iteration number Thenumerical solution of (9) did not exhibit convergent problemthus Jacobi iteration method was sufficient
Table 1 The coefficient values from Haverkamp et al (1977) [7]based on (10) These values were used as base case Note that 120579
119903is
residual volumetric water content 120579119904is saturated volumetric water
content 119870119904is saturated hydraulic conductivity and 120572 120573 119860 and 119861
are fitting coefficients
Parameter Value120572 739120579119903
0124m3mminus3
120579119904
0495m3mminus3
120573 4119860 1246119861 177119870119904
123 times 10minus7msminus1
00
50
100
150
200
250
300
020 025 030 035 040 045 050
Simulated 105 s
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Philip(H) at 105 sPhilip(K) at 105 s
Figure 1 Comparison of simulated results with Philiprsquos semianalyt-ical solution Philip(H) and Philip(K) were from Haverkamp et al[7] and Kabala and Milly [8] respectively
24 Statistical Measures In order to determine the goodnessof fit between reference data and simulated results onestatistical equation was implemented The equation is calledabsolute residual errors (MA) as follows [12]
MA =1
119873
119873
sum
119896=1
1003816100381610038161003816cal119896 minus obs119896
1003816100381610038161003816 (11)
where cal119896is the simulated data at cell 119896 and obs
119896is the
analytical solution as reference data at cell 119896
3 Results and Discussion
31 Simulation Results and Their Accuracy Based on theconditions as stated in previous section water infiltration intoYolo light clay was simulated up to 105 s Data on Philiprsquossemianalytical solution were collected fromHaverkamp et al[7] hereafter referred to as Philip(H) Simulation results werecompared with the data to verify the simulation (Figure 1)It was evident that the simulation results slightly underpre-dicted the infiltration front of water flow
To further reinforce the previous claim some datawere extracted from Kabala and Milly [8] as indicated by
4 International Scholarly Research Notices
Philip(K) as in Figure 1 for further comparison Figure 1shows that there was a small difference between Philip(K)and Philip(H) but the former was relatively closer to thesimulation results than the latter At this point of observationwe were unable to determine which of the solutions thatis Philip(K) and Philip(H) provided from the literature wasaccurateHowever results from the figure clearly indicate thatthe simulated result was lesser than Philiprsquos semianalyticalsolution Therefore sensitivity analysis was carried out todetermine the sensitivity coefficient for all input parametersand use the sensitivity analysis results to assess the modelsimulation based on the assumption that possibly the cumu-lative effect of input parameters in terms of significant digitsapproximation could be contributing to the underpredictionof the volumetric water content of the simulation In additionsensitivity analysis is one of the most important steps inevaluating the effect of input parameter on simulation resultsand it is also used by others for model validation [13ndash16]
32 Sensitivity Analysis and Simulation Model ValidationNegligible sensitivity response could be due to too small per-turbation size and inaccuracy in sensitivity response couldbe due to too large perturbation size [17] Values of inputparameters were subjected to a perturbation size betweenminus5 and 5 as suggested by Zheng and Bennett [12] andin considering the simulation time we limit the sensitivityanalysis to a simulation time of 105 s The sensitivity analysisstudy was based on a single perturbation size of incrementor decrement in each simulation The sensitivity analysiswas carried out based on the hydraulic functions (10) fromHaverkamp et al [7]
There were seven input parameters from Haverkamphydraulic functions as listed in Table 1 Additional fourinput parameters were also tested that is initial volumetricwater content (120579
119871(initial cond)) boundary volumetric water
content 120579119871(upper cond) time-step size (Δ119905) and spatial
spacing size (Δ119911) The depth at 155 cm from the groundsurface was used for observation
The normalized sensitivity coefficients are shown inFigure 2 Generally there are two groups of sensitivity coef-ficients that is positive and negative relations In positiverelation group the boundary volumetric water content hadthe highest sensitivity coefficientThis was followed by initialvolumetric water content and saturated hydraulic conductiv-ity The smallest sensitivity coefficient in the group was theresidual volumetric water content In negative relation groupsaturated volumetric water content had the highest sensitivitycoefficient and this group endedwith spatial spacing size andtime-step size as the smallest sensitivity coefficient
For comparison purpose elementary effects method wasalso used to calculate normalized sensitivity coefficient Weassumed only random generation in 119896-dimensional diagonalmatrix (Dlowast) and then (2) was used to generate 50 trajectoriesEquation (3) was used to screen out 4 trajectories with thegreatest geometric distance of those trajectories Equations(4) to (7) were used to calculate the elementary effects meanof elementary effects mean of absolute values of the elemen-tary effects and standard deviation respectivelyThemean ofelementary effects was modified to calculate the normalized
100300500700900
1100
Input parameter
minus100
minus300
minus500
minus700
minus900
minus1100
Nor
mal
ized
sens
itivi
ty co
effici
ent (
)
120579L
(upp
er b
ound
)
120579L
(initi
al co
nd)120579s 120579r
Ks120573 A120572Δz
ΔtB
644Eminus01
508E
minus01
208Eminus01
825Eminus02
385Eminus02
minus619Eminus04
minus550E
minus02
minus589Eminus01
minus107E+00
minus497E+00
471E
+00
Figure 2 The rank of sensitivity coefficient Note 120579119904and 120579
119903
are saturated and residual volumetric water content Δ119911 spatialspacing sizeΔ119905 time-step size119870
119904 saturated hydraulic conductivity
120579119871(initial cond) clay medium initial value of volumetric water
content 120579119871(upper bound) upper boundary of volumetric water
content 119860 119861 120573 and 120572 are fitting parameters from Haverkamp asin (10)
Table 2 Statistical measures (120583 120583lowast and 120590) of elementary effectsmethod They are the mean of elementary effects the meanof absolute values of the elementary effects and the standarddeviation respectively Note that 120579
119903is residual volumetric water
content 120579119904is saturated volumetric water content 119870
119904is saturated
hydraulic conductivityΔ119911 is spatial spacing sizeΔ119905 is time-step size120579119871(initial cond) is initial value of volumetric water content and 120572
120573 119860 and 119861 are fitting coefficients
120583 () 120583lowast () 120590
120579119904
minus603119864 + 00 603119864 + 00 948119864 minus 01
119861 minus185119864 + 00 185119864 + 00 938119864 minus 01
120573 minus207119864 minus 01 320119864 minus 01 370119864 minus 01
120572 minus414119864 minus 02 125119864 minus 01 147119864 minus 01
Δ119911 minus335119864 minus 02 395119864 minus 02 456119864 minus 02
Δ119905 minus166119864 minus 04 525119864 minus 04 621119864 minus 04
120579119903
444119864 minus 03 337119864 minus 02 402119864 minus 02
119860 313119864 minus 01 313119864 minus 01 741119864 minus 02
119870119904
524119864 minus 01 524119864 minus 01 673119864 minus 02
120579119871(initial cond) 884119864 minus 01 884119864 minus 01 305119864 minus 01
sensitivity coefficient The results are shown in Table 2 Thesensitivity coefficient has identical ranking as those obtainedin Figure 2 except for the coefficient of 120572 input parameterSimilar values of 120583 and 120583
lowast indicate linear effect on few inputparameters in positive (119860119870
119904 and 120579
119871(initial cond)) and nega-
tive (120579119904and 119861) relations Other input parameters have shown
the effect of oscillating sign that results in different values of120583 and 120583
lowast In general those sensitivity coefficients generatedby different methods have shown comparable results
We assumed that a minor deviation in each input param-eter in terms of its significant digits approximation couldcontribute some effects on the simulation outcome thatcould possibly explain the discrepancy between the simulatedresults and Philiprsquos semianalytical solution (Figure 1) In other
International Scholarly Research Notices 5
Table 3 Significant digits approximation on input parameter valueNote that 120579
119903is residual volumetric water content 120579
119904is saturated
volumetric water content 119870119904is saturated hydraulic conductivity
Δ119911 is spatial spacing size Δ119905 is time-step size 120579119871(initial cond) is
initial value of volumetric water content 120579119871(upper bond) is upper
boundary of volumetric water content and 120572 120573119860 and 119861 are fittingcoefficients
Δ119905 10 s the base case was 500 sΔ119911 01 cm the base case was 1 cm
words the parameter values in terms of significant digitsapproximation that were used in computer simulation byHaverkamp et al [7] could be different from the exactdata in terms of input parameter significant digits thatthey published Thus we take advantage on the positive andnegative relations generated from the sensitivity analysis andset up a hypothetical approximation value in Table 3 forfurther investigation The cumulative effect was studied bymanipulating an input parameter used for each simulationand the subsequent manipulation of input parameter wascarried out on top of the previous changed input parameterThis process begins from step 1 for base case to step 10 forspatial spacing size For instance the 120579
119871(initial cond) value
(02376499m3mminus3) was used as a second simulation (in step2) after the base case simulation This was followed by thirdsimulation (in step 3) using 120579
119903value as 0124499m3mminus3
by remaining 120579119871(initial cond) value used in the second
simulation For each simulation Equation (11) was used tocalculate the discrepancy between simulation results andPhiliprsquos semianalytical solution (data from [7]) for absoluteresidual error (MA)Of all those eleven parameters inTable 3Δ119905 and Δ119911 were the only two parameters without any limit ofvariation and for this reason we extend the variation limitby reducing the former and the latter by 98 and 90 from500 s and 1 cm to 10 s and 01 cm respectively The 120579
119904and
120579119871(119906119901119901119890119903 119887119900119906119899119889) values are negative and positive relations
respectively Decreasing and increasing the correspondingformer and latter values would result in simulation failurethus those two parameters remained unchanged
A consistent reduction inMA value from 120579119871(initial cond)
to 119861 input parameter was observed except a slight incrementat Δ119905 input parameter simulation and a steep slide of MAvalue was observed on the Δ119911 input parameter simulation(Figure 3) Although the sensitivity coefficient in Figure 2indicates that reducing Δ119905 value should lead to a reductionin MA value the simulated result showed an increase in
002
54
002
53
002
53
002
45
002
45
002
45
002
45
002
338
002
343
001
06
0010
0015
0020
0025
0030
0035
Input parameter
Abso
lute
resid
ual e
rror
(MA
)
Δz
(ste
p10
)
Δt
(ste
p9
)
B(s
tep8
)
A(s
tep7
)
Ks
(ste
p6
)
120572(s
tep5
)
120573(s
tep4
)
120579L
(initi
al co
nd)
(ste
p2
)
Base
case
(ste
p 1)
120579r
(ste
p3
)
(m3
mminus3)
Figure 3The cumulative effect of input parameters on the absoluteresidual error at simulation time 105 s Note 120579
119871(initial cond) (step
2) claymedium initial value of volumetric water content 120579119903(step 3)
residual volumetric water content 120573 (step 4) 120572 (step 5) 119860 (step 7)and 119861 (step 8) are fitting parameters119870
the MA value This observation could be explained from theresult of elementary effects method This was because Δ119905 hasdifferent values of 120583 and 120583
lowast which indicate the capability ofsign oscillation (Table 2)
Figure 3 shows that the simulation on the cumulativeeffect of steps 2ndash9 which combined the effect from 120579
119871(initial
cond) (step 2) with Δ119905 (step 9) did not contribute to anysignificant effects on the advancement of water infiltrationfront It only resulted in a reduction of 78 in MA valuefrom 00254 to 002343m3mminus3 In addition those eight inputparameters had to vary in significant digits approximation astabulated in Table 3 in order to result in the stated percentagereduction Therefore the significant digits approximationmight not be the main cause of the problem in consideringthat a more significant effect on the advance of water infiltra-tion front was shown by Δ119911 in the Figure 4 A further step toinclude Δ119911 in the simulation that is the cumulative effect ofsteps 2ndash10 which combined the effect from 120579
119871(initial cond)
(step 2) with Δ119911 (step 10) there was 547 reduction in MAvalue of step 9 from 002343 to 00106m3mminus3 This indicatesthat the spatial spacing size was themain cause in the advanceof water infiltration front Therefore the simulation wasrepeated for the last time for the effect of spatial spacing sizealone and in Figure 4 there was a good agreement betweenthe simulation results and the Philip(K) This observationcould be explained using (1) after rearranging it into thefollowing form which we termed as percentage variation insimulation results
Δ119910119894
119910119894
=Δ119886119908
119886119908
119883119894119908 (12)
6 International Scholarly Research Notices
16
17
18
19
20
21
22
23
027 029 031 033 035 037 039
Philip(H)
Base case Philip(K)Only step 10
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Steps 2ndash10Steps 2ndash9
Figure 4 The effect of Δ119911 (step 10 alone) and cumulative effects ofsteps 2 to 9 and 2ndash10 in comparison with Philip(H) and Philip(K)
where Δ119886119908119886119908
is the normalized input parameter valueΔ119910119894119910119894is the normalized output parameter value and 119883
119894119908
is the normalized sensitivity coefficient () Equation (12)is simply a multiplication of the percentage change in inputparameter value from the base case and the normalizedsensitivity coefficient
Using (12) the percentage variation in simulation resultsfrom input parameters of Δ119911 and Δ119905 caused an incrementof 495 and 006 respectively despite Δ119905 having thehighest reduction in percentage (minus98) from base case Thisobservation could be summarized as follows firstly inputparameter with the highest sensitivity coefficient does notguarantee the greatest effect on the simulation result forexample 120579
119871(initial cond) secondly input parameter with
the highest percentage of change also does not guaranteethe greatest effect on the simulation result for exampleΔ119905 and therefore only the highest sensitivity coefficientwith the highest percentage change on input parameter (orthe uncertainty) would give the most substantial effect onsimulation result
4 Conclusions
Thegoverning equation of transientwater flow in unsaturatedand nonisothermal conditions was approximated numeri-cally by finite-difference solution It was successfully imple-mented into FORTRAN programming language simulatedand verified by Philiprsquos semianalytical solution on waterinfiltration into Yolo light clay with data from literatures
One-at-a-timeOAT and elementary effects EEmethodswere used in the sensitivity analysis A common trend ofsensitivity was observed across the methods in both positiveand negative relationsThe latter method allowed explorationof additional characteristics of input parameters at differentinput space such as linearity and sign oscillation effectThe sign oscillation effect observed on input parametersexplained the possibility of its deviation from those observedin OAT method at different input spaces
A hypothetical case that was established to study thecumulative effect of input parameters on the discrepancybetween simulated result and Philiprsquos semianalytical solutionin terms of significant digits approximation (from base case)was found to be unlikely A large normalized sensitivitycoefficient was made with initial volumetric water contentand the largest percentage changes were with time-step sizebut surprisingly none of them contributes to any substantialimpact on simulation results when compared to spatialspacing size This observation led to the conclusion that theuncertainty of input parameter and normalized sensitivitycoefficient of input parameters both controlled the outcomeof simulation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportfrom Meiji University Japan and the Ministry of EducationMalaysia and alsoUniversitiMalaysia TerengganuMalaysia
References
[1] D J Pannell ldquoSensitivity analysis of normative economic mod-els theoretical framework and practical strategiesrdquoAgriculturalEconomics vol 16 no 2 pp 139ndash152 1997
[2] D M Hamby ldquoA review of techniques for parameter sensitivityanalysis of environmental modelsrdquo Environmental Monitoringand Assessment vol 32 no 2 pp 135ndash154 1994
[3] A Saltelli S Tarantola F Campolongo andMRatto SensitivityAnalysis in Practice A Guide to Assessing Scientific Models JohnWiley amp Sons Hoboken NJ USA 2004
[4] A Saltelli and P Annoni ldquoHow to avoid a perfunctory sensitiv-ity analysisrdquo Environmental Modelling and Software vol 25 no12 pp 1508ndash1517 2010
[5] A Saltelli M Ratto and T Andres Global Sensitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008
[6] F Campolongo J Cariboni and A Saltelli ldquoAn effectivescreening design for sensitivity analysis of large modelsrdquo Envi-ronmental Modelling and Software vol 22 no 10 pp 1509ndash15182007
[7] R Haverkamp M Vauclin J Touma P J Wierenga and GVachaud ldquoA comparison of numerical simulation models forone-dimensional infiltrationrdquo Soil Science Society of AmericaJournal vol 41 no 2 pp 285ndash294 1977
[8] Z J Kabala and P C D Milly ldquoSensitivity analysis of flowin unsaturated heterogeneous porous media theory numericalmodel and its verificationrdquo Water Resources Research vol 26no 4 pp 593ndash610 1990
[9] M A Celia E T Bouloutas and R L Zarba ldquoA generalmass-conservative numerical solution for the unsaturated flowequationrdquo Water Resources Research vol 26 no 7 pp 1483ndash1496 1990
[10] J Tu G H Yeoh and C Liu Computational Fluid DynamicsA Practical Approach Butterworth-Heinemann Oxford UK2008
International Scholarly Research Notices 7
[11] J Istok ldquoStep 4 solve system of equationsrdquo in GroundwaterModeling by the Finite Element Method chapter 5 pp 176ndash225American Geophysical Union Washington DC USA 1989
[12] C Zheng and G D Bennett Applied Contaminant TransportModeling JohnWiley amp Sons New York NY USA 2nd edition2002
[13] F Stange K Butterbach-Bahl H Papen S Zechmeister-Boltenstern C Li and J Aber ldquoA process-oriented model ofN2O and NO emissions from forest soils 2 Sensitivity analysis
and validationrdquo Journal of Geophysical Research Atmospheresvol 105 no 4 Article ID 1999JD900948 pp 4385ndash4398 2000
[14] R Nathan U N Safriel and I Noy-Meir ldquoField validation andsensitivity analysis of a mechanistic model for tree seed dis-persal by windrdquo Ecology vol 82 no 2 pp 374ndash388 2001
[15] C HMin Y L He X L Liu B H YinW Jiang andW Q TaoldquoParameter sensitivity examination and discussion of PEM fuelcell simulation model validation Part II Results of sensitivityanalysis and validation of the modelrdquo Journal of Power Sourcesvol 160 no 1 pp 374ndash385 2006
[16] S N Gosling and N W Arnell ldquoSimulating current globalriver runoff with a global hydrological model model revisionsvalidation and sensitivity analysisrdquo Hydrological Processes vol25 no 7 pp 1129ndash1145 2011
[17] E P Poeter and M C Hill Documentation of UCODE A Com-puter Code for Universal Inverse Modeling vol 98 DIANEPublishing 1998
where 119896 indicates a cell-centered number in 119911-directionin Cartesian coordinate system Δ119905 (s) is time-step size120579119871(119896)
119899 (m3mminus3) and 120579119871(119896)
119899+1 (m3mminus3) are volumetric watercontent at old time level (119899) and new time level (119899 + 1)respectively 119870
119896+12(m sminus1) is hydraulic conductivity at the
interface between cells 119896 and 119896+1119870119896minus12
(m sminus1) is hydraulicconductivity at the interface between cells 119896 minus 1 and 119896(120597120595119898120597120579119871)119896+12
is partial derivative of 120595119898with respect to 120579
119871
at the interface between cells 119896 and 119896 + 1 (120597120595119898120597120579119871)119896minus12
ispartial derivative of 120595
119898with respect to 120579
119871at the interface
between cells 119896 minus 1 and 119896 Δ119911119896+1
(m) Δ119911119896(m) and Δ119911
119896minus1(m)
are corresponding to the spatial sizes of spacing of cells 119896+1 119896and 119896minus1 respectively 120579
119871(119896+1)
119899+1 (m3mminus3) 120579119871(119896)
119899+1 (m3mminus3)and 120579
119871(119896minus1)
119899+1 (m3mminus3) are the volumetric water contents atnew time level of cells 119896+1 119896 and 119896minus1 respectively Equation(8) was numerically solved by a fully implicit cell-centeredfinite-difference scheme without any linearization An itera-tive method was used to solve the mathematical algebra of(9) that is Jacobi iteration [10] For comparison purposemodified Newton-Raphson method was also implemented[11] A convergence factor criterion was used to indicate thecondition for iteration termination process that is absolutemaximum difference |120579
119871(119896)
119899+1minus 120579119871(119896)
119899| for every single cell
22 The Constitutive Functions of Matric Pressure Head (120595119898)
and Hydraulic Conductivity (119870) The hydraulic functionsused were adopted from Haverkamp et al [7]
where 120572 120573 119860 and 119861 are fitting parameters 120579119903(m3mminus3)
is residual volumetric water content 120579119904(m3mminus3) is satu-
rated volumetric water content and 119870119904(m sminus1) is saturated
hydraulic conductivity
23 Numerical Experiment and the Default Setting of InputParameters of the Flow Problem Water infiltration intoYolo light clay was used in the numerical experiment Thehydraulic functions for the soil (see (10)) and the coefficientsvalues are shown in Table 1 Initial condition for the volumet-ric water content was 02376m3mminus3 Lower boundary wasset as free-drainage to water flow Upper boundary was setat 0495m3mminus3 After considering the mass balance ratio [9]and iteration number the time-step size spatial discretizationsize and convergent value were set at corresponding valuesof 500 s 1 cm and 10minus12m3mminus3 respectively The iterationmethods of Jacobi and modified Newton-Raphson werecompared It was found that the minimum iteration numberfrom the latter was equivalent to the iteration number fromthe former when the relaxation factor of the latter was setto unity (data not shown) Reducing the relaxation factorfrom unity would result in increasing iteration number Thenumerical solution of (9) did not exhibit convergent problemthus Jacobi iteration method was sufficient
Table 1 The coefficient values from Haverkamp et al (1977) [7]based on (10) These values were used as base case Note that 120579
119903is
residual volumetric water content 120579119904is saturated volumetric water
content 119870119904is saturated hydraulic conductivity and 120572 120573 119860 and 119861
are fitting coefficients
Parameter Value120572 739120579119903
0124m3mminus3
120579119904
0495m3mminus3
120573 4119860 1246119861 177119870119904
123 times 10minus7msminus1
00
50
100
150
200
250
300
020 025 030 035 040 045 050
Simulated 105 s
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Philip(H) at 105 sPhilip(K) at 105 s
Figure 1 Comparison of simulated results with Philiprsquos semianalyt-ical solution Philip(H) and Philip(K) were from Haverkamp et al[7] and Kabala and Milly [8] respectively
24 Statistical Measures In order to determine the goodnessof fit between reference data and simulated results onestatistical equation was implemented The equation is calledabsolute residual errors (MA) as follows [12]
MA =1
119873
119873
sum
119896=1
1003816100381610038161003816cal119896 minus obs119896
1003816100381610038161003816 (11)
where cal119896is the simulated data at cell 119896 and obs
119896is the
analytical solution as reference data at cell 119896
3 Results and Discussion
31 Simulation Results and Their Accuracy Based on theconditions as stated in previous section water infiltration intoYolo light clay was simulated up to 105 s Data on Philiprsquossemianalytical solution were collected fromHaverkamp et al[7] hereafter referred to as Philip(H) Simulation results werecompared with the data to verify the simulation (Figure 1)It was evident that the simulation results slightly underpre-dicted the infiltration front of water flow
To further reinforce the previous claim some datawere extracted from Kabala and Milly [8] as indicated by
4 International Scholarly Research Notices
Philip(K) as in Figure 1 for further comparison Figure 1shows that there was a small difference between Philip(K)and Philip(H) but the former was relatively closer to thesimulation results than the latter At this point of observationwe were unable to determine which of the solutions thatis Philip(K) and Philip(H) provided from the literature wasaccurateHowever results from the figure clearly indicate thatthe simulated result was lesser than Philiprsquos semianalyticalsolution Therefore sensitivity analysis was carried out todetermine the sensitivity coefficient for all input parametersand use the sensitivity analysis results to assess the modelsimulation based on the assumption that possibly the cumu-lative effect of input parameters in terms of significant digitsapproximation could be contributing to the underpredictionof the volumetric water content of the simulation In additionsensitivity analysis is one of the most important steps inevaluating the effect of input parameter on simulation resultsand it is also used by others for model validation [13ndash16]
32 Sensitivity Analysis and Simulation Model ValidationNegligible sensitivity response could be due to too small per-turbation size and inaccuracy in sensitivity response couldbe due to too large perturbation size [17] Values of inputparameters were subjected to a perturbation size betweenminus5 and 5 as suggested by Zheng and Bennett [12] andin considering the simulation time we limit the sensitivityanalysis to a simulation time of 105 s The sensitivity analysisstudy was based on a single perturbation size of incrementor decrement in each simulation The sensitivity analysiswas carried out based on the hydraulic functions (10) fromHaverkamp et al [7]
There were seven input parameters from Haverkamphydraulic functions as listed in Table 1 Additional fourinput parameters were also tested that is initial volumetricwater content (120579
119871(initial cond)) boundary volumetric water
content 120579119871(upper cond) time-step size (Δ119905) and spatial
spacing size (Δ119911) The depth at 155 cm from the groundsurface was used for observation
The normalized sensitivity coefficients are shown inFigure 2 Generally there are two groups of sensitivity coef-ficients that is positive and negative relations In positiverelation group the boundary volumetric water content hadthe highest sensitivity coefficientThis was followed by initialvolumetric water content and saturated hydraulic conductiv-ity The smallest sensitivity coefficient in the group was theresidual volumetric water content In negative relation groupsaturated volumetric water content had the highest sensitivitycoefficient and this group endedwith spatial spacing size andtime-step size as the smallest sensitivity coefficient
For comparison purpose elementary effects method wasalso used to calculate normalized sensitivity coefficient Weassumed only random generation in 119896-dimensional diagonalmatrix (Dlowast) and then (2) was used to generate 50 trajectoriesEquation (3) was used to screen out 4 trajectories with thegreatest geometric distance of those trajectories Equations(4) to (7) were used to calculate the elementary effects meanof elementary effects mean of absolute values of the elemen-tary effects and standard deviation respectivelyThemean ofelementary effects was modified to calculate the normalized
100300500700900
1100
Input parameter
minus100
minus300
minus500
minus700
minus900
minus1100
Nor
mal
ized
sens
itivi
ty co
effici
ent (
)
120579L
(upp
er b
ound
)
120579L
(initi
al co
nd)120579s 120579r
Ks120573 A120572Δz
ΔtB
644Eminus01
508E
minus01
208Eminus01
825Eminus02
385Eminus02
minus619Eminus04
minus550E
minus02
minus589Eminus01
minus107E+00
minus497E+00
471E
+00
Figure 2 The rank of sensitivity coefficient Note 120579119904and 120579
119903
are saturated and residual volumetric water content Δ119911 spatialspacing sizeΔ119905 time-step size119870
119904 saturated hydraulic conductivity
120579119871(initial cond) clay medium initial value of volumetric water
content 120579119871(upper bound) upper boundary of volumetric water
content 119860 119861 120573 and 120572 are fitting parameters from Haverkamp asin (10)
Table 2 Statistical measures (120583 120583lowast and 120590) of elementary effectsmethod They are the mean of elementary effects the meanof absolute values of the elementary effects and the standarddeviation respectively Note that 120579
119903is residual volumetric water
content 120579119904is saturated volumetric water content 119870
119904is saturated
hydraulic conductivityΔ119911 is spatial spacing sizeΔ119905 is time-step size120579119871(initial cond) is initial value of volumetric water content and 120572
120573 119860 and 119861 are fitting coefficients
120583 () 120583lowast () 120590
120579119904
minus603119864 + 00 603119864 + 00 948119864 minus 01
119861 minus185119864 + 00 185119864 + 00 938119864 minus 01
120573 minus207119864 minus 01 320119864 minus 01 370119864 minus 01
120572 minus414119864 minus 02 125119864 minus 01 147119864 minus 01
Δ119911 minus335119864 minus 02 395119864 minus 02 456119864 minus 02
Δ119905 minus166119864 minus 04 525119864 minus 04 621119864 minus 04
120579119903
444119864 minus 03 337119864 minus 02 402119864 minus 02
119860 313119864 minus 01 313119864 minus 01 741119864 minus 02
119870119904
524119864 minus 01 524119864 minus 01 673119864 minus 02
120579119871(initial cond) 884119864 minus 01 884119864 minus 01 305119864 minus 01
sensitivity coefficient The results are shown in Table 2 Thesensitivity coefficient has identical ranking as those obtainedin Figure 2 except for the coefficient of 120572 input parameterSimilar values of 120583 and 120583
lowast indicate linear effect on few inputparameters in positive (119860119870
119904 and 120579
119871(initial cond)) and nega-
tive (120579119904and 119861) relations Other input parameters have shown
the effect of oscillating sign that results in different values of120583 and 120583
lowast In general those sensitivity coefficients generatedby different methods have shown comparable results
We assumed that a minor deviation in each input param-eter in terms of its significant digits approximation couldcontribute some effects on the simulation outcome thatcould possibly explain the discrepancy between the simulatedresults and Philiprsquos semianalytical solution (Figure 1) In other
International Scholarly Research Notices 5
Table 3 Significant digits approximation on input parameter valueNote that 120579
119903is residual volumetric water content 120579
119904is saturated
volumetric water content 119870119904is saturated hydraulic conductivity
Δ119911 is spatial spacing size Δ119905 is time-step size 120579119871(initial cond) is
initial value of volumetric water content 120579119871(upper bond) is upper
boundary of volumetric water content and 120572 120573119860 and 119861 are fittingcoefficients
Δ119905 10 s the base case was 500 sΔ119911 01 cm the base case was 1 cm
words the parameter values in terms of significant digitsapproximation that were used in computer simulation byHaverkamp et al [7] could be different from the exactdata in terms of input parameter significant digits thatthey published Thus we take advantage on the positive andnegative relations generated from the sensitivity analysis andset up a hypothetical approximation value in Table 3 forfurther investigation The cumulative effect was studied bymanipulating an input parameter used for each simulationand the subsequent manipulation of input parameter wascarried out on top of the previous changed input parameterThis process begins from step 1 for base case to step 10 forspatial spacing size For instance the 120579
119871(initial cond) value
(02376499m3mminus3) was used as a second simulation (in step2) after the base case simulation This was followed by thirdsimulation (in step 3) using 120579
119903value as 0124499m3mminus3
by remaining 120579119871(initial cond) value used in the second
simulation For each simulation Equation (11) was used tocalculate the discrepancy between simulation results andPhiliprsquos semianalytical solution (data from [7]) for absoluteresidual error (MA)Of all those eleven parameters inTable 3Δ119905 and Δ119911 were the only two parameters without any limit ofvariation and for this reason we extend the variation limitby reducing the former and the latter by 98 and 90 from500 s and 1 cm to 10 s and 01 cm respectively The 120579
119904and
120579119871(119906119901119901119890119903 119887119900119906119899119889) values are negative and positive relations
respectively Decreasing and increasing the correspondingformer and latter values would result in simulation failurethus those two parameters remained unchanged
A consistent reduction inMA value from 120579119871(initial cond)
to 119861 input parameter was observed except a slight incrementat Δ119905 input parameter simulation and a steep slide of MAvalue was observed on the Δ119911 input parameter simulation(Figure 3) Although the sensitivity coefficient in Figure 2indicates that reducing Δ119905 value should lead to a reductionin MA value the simulated result showed an increase in
002
54
002
53
002
53
002
45
002
45
002
45
002
45
002
338
002
343
001
06
0010
0015
0020
0025
0030
0035
Input parameter
Abso
lute
resid
ual e
rror
(MA
)
Δz
(ste
p10
)
Δt
(ste
p9
)
B(s
tep8
)
A(s
tep7
)
Ks
(ste
p6
)
120572(s
tep5
)
120573(s
tep4
)
120579L
(initi
al co
nd)
(ste
p2
)
Base
case
(ste
p 1)
120579r
(ste
p3
)
(m3
mminus3)
Figure 3The cumulative effect of input parameters on the absoluteresidual error at simulation time 105 s Note 120579
119871(initial cond) (step
2) claymedium initial value of volumetric water content 120579119903(step 3)
residual volumetric water content 120573 (step 4) 120572 (step 5) 119860 (step 7)and 119861 (step 8) are fitting parameters119870
the MA value This observation could be explained from theresult of elementary effects method This was because Δ119905 hasdifferent values of 120583 and 120583
lowast which indicate the capability ofsign oscillation (Table 2)
Figure 3 shows that the simulation on the cumulativeeffect of steps 2ndash9 which combined the effect from 120579
119871(initial
cond) (step 2) with Δ119905 (step 9) did not contribute to anysignificant effects on the advancement of water infiltrationfront It only resulted in a reduction of 78 in MA valuefrom 00254 to 002343m3mminus3 In addition those eight inputparameters had to vary in significant digits approximation astabulated in Table 3 in order to result in the stated percentagereduction Therefore the significant digits approximationmight not be the main cause of the problem in consideringthat a more significant effect on the advance of water infiltra-tion front was shown by Δ119911 in the Figure 4 A further step toinclude Δ119911 in the simulation that is the cumulative effect ofsteps 2ndash10 which combined the effect from 120579
119871(initial cond)
(step 2) with Δ119911 (step 10) there was 547 reduction in MAvalue of step 9 from 002343 to 00106m3mminus3 This indicatesthat the spatial spacing size was themain cause in the advanceof water infiltration front Therefore the simulation wasrepeated for the last time for the effect of spatial spacing sizealone and in Figure 4 there was a good agreement betweenthe simulation results and the Philip(K) This observationcould be explained using (1) after rearranging it into thefollowing form which we termed as percentage variation insimulation results
Δ119910119894
119910119894
=Δ119886119908
119886119908
119883119894119908 (12)
6 International Scholarly Research Notices
16
17
18
19
20
21
22
23
027 029 031 033 035 037 039
Philip(H)
Base case Philip(K)Only step 10
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Steps 2ndash10Steps 2ndash9
Figure 4 The effect of Δ119911 (step 10 alone) and cumulative effects ofsteps 2 to 9 and 2ndash10 in comparison with Philip(H) and Philip(K)
where Δ119886119908119886119908
is the normalized input parameter valueΔ119910119894119910119894is the normalized output parameter value and 119883
119894119908
is the normalized sensitivity coefficient () Equation (12)is simply a multiplication of the percentage change in inputparameter value from the base case and the normalizedsensitivity coefficient
Using (12) the percentage variation in simulation resultsfrom input parameters of Δ119911 and Δ119905 caused an incrementof 495 and 006 respectively despite Δ119905 having thehighest reduction in percentage (minus98) from base case Thisobservation could be summarized as follows firstly inputparameter with the highest sensitivity coefficient does notguarantee the greatest effect on the simulation result forexample 120579
119871(initial cond) secondly input parameter with
the highest percentage of change also does not guaranteethe greatest effect on the simulation result for exampleΔ119905 and therefore only the highest sensitivity coefficientwith the highest percentage change on input parameter (orthe uncertainty) would give the most substantial effect onsimulation result
4 Conclusions
Thegoverning equation of transientwater flow in unsaturatedand nonisothermal conditions was approximated numeri-cally by finite-difference solution It was successfully imple-mented into FORTRAN programming language simulatedand verified by Philiprsquos semianalytical solution on waterinfiltration into Yolo light clay with data from literatures
One-at-a-timeOAT and elementary effects EEmethodswere used in the sensitivity analysis A common trend ofsensitivity was observed across the methods in both positiveand negative relationsThe latter method allowed explorationof additional characteristics of input parameters at differentinput space such as linearity and sign oscillation effectThe sign oscillation effect observed on input parametersexplained the possibility of its deviation from those observedin OAT method at different input spaces
A hypothetical case that was established to study thecumulative effect of input parameters on the discrepancybetween simulated result and Philiprsquos semianalytical solutionin terms of significant digits approximation (from base case)was found to be unlikely A large normalized sensitivitycoefficient was made with initial volumetric water contentand the largest percentage changes were with time-step sizebut surprisingly none of them contributes to any substantialimpact on simulation results when compared to spatialspacing size This observation led to the conclusion that theuncertainty of input parameter and normalized sensitivitycoefficient of input parameters both controlled the outcomeof simulation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportfrom Meiji University Japan and the Ministry of EducationMalaysia and alsoUniversitiMalaysia TerengganuMalaysia
References
[1] D J Pannell ldquoSensitivity analysis of normative economic mod-els theoretical framework and practical strategiesrdquoAgriculturalEconomics vol 16 no 2 pp 139ndash152 1997
[2] D M Hamby ldquoA review of techniques for parameter sensitivityanalysis of environmental modelsrdquo Environmental Monitoringand Assessment vol 32 no 2 pp 135ndash154 1994
[3] A Saltelli S Tarantola F Campolongo andMRatto SensitivityAnalysis in Practice A Guide to Assessing Scientific Models JohnWiley amp Sons Hoboken NJ USA 2004
[4] A Saltelli and P Annoni ldquoHow to avoid a perfunctory sensitiv-ity analysisrdquo Environmental Modelling and Software vol 25 no12 pp 1508ndash1517 2010
[5] A Saltelli M Ratto and T Andres Global Sensitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008
[6] F Campolongo J Cariboni and A Saltelli ldquoAn effectivescreening design for sensitivity analysis of large modelsrdquo Envi-ronmental Modelling and Software vol 22 no 10 pp 1509ndash15182007
[7] R Haverkamp M Vauclin J Touma P J Wierenga and GVachaud ldquoA comparison of numerical simulation models forone-dimensional infiltrationrdquo Soil Science Society of AmericaJournal vol 41 no 2 pp 285ndash294 1977
[8] Z J Kabala and P C D Milly ldquoSensitivity analysis of flowin unsaturated heterogeneous porous media theory numericalmodel and its verificationrdquo Water Resources Research vol 26no 4 pp 593ndash610 1990
[9] M A Celia E T Bouloutas and R L Zarba ldquoA generalmass-conservative numerical solution for the unsaturated flowequationrdquo Water Resources Research vol 26 no 7 pp 1483ndash1496 1990
[10] J Tu G H Yeoh and C Liu Computational Fluid DynamicsA Practical Approach Butterworth-Heinemann Oxford UK2008
International Scholarly Research Notices 7
[11] J Istok ldquoStep 4 solve system of equationsrdquo in GroundwaterModeling by the Finite Element Method chapter 5 pp 176ndash225American Geophysical Union Washington DC USA 1989
[12] C Zheng and G D Bennett Applied Contaminant TransportModeling JohnWiley amp Sons New York NY USA 2nd edition2002
[13] F Stange K Butterbach-Bahl H Papen S Zechmeister-Boltenstern C Li and J Aber ldquoA process-oriented model ofN2O and NO emissions from forest soils 2 Sensitivity analysis
and validationrdquo Journal of Geophysical Research Atmospheresvol 105 no 4 Article ID 1999JD900948 pp 4385ndash4398 2000
[14] R Nathan U N Safriel and I Noy-Meir ldquoField validation andsensitivity analysis of a mechanistic model for tree seed dis-persal by windrdquo Ecology vol 82 no 2 pp 374ndash388 2001
[15] C HMin Y L He X L Liu B H YinW Jiang andW Q TaoldquoParameter sensitivity examination and discussion of PEM fuelcell simulation model validation Part II Results of sensitivityanalysis and validation of the modelrdquo Journal of Power Sourcesvol 160 no 1 pp 374ndash385 2006
[16] S N Gosling and N W Arnell ldquoSimulating current globalriver runoff with a global hydrological model model revisionsvalidation and sensitivity analysisrdquo Hydrological Processes vol25 no 7 pp 1129ndash1145 2011
[17] E P Poeter and M C Hill Documentation of UCODE A Com-puter Code for Universal Inverse Modeling vol 98 DIANEPublishing 1998
Philip(K) as in Figure 1 for further comparison Figure 1shows that there was a small difference between Philip(K)and Philip(H) but the former was relatively closer to thesimulation results than the latter At this point of observationwe were unable to determine which of the solutions thatis Philip(K) and Philip(H) provided from the literature wasaccurateHowever results from the figure clearly indicate thatthe simulated result was lesser than Philiprsquos semianalyticalsolution Therefore sensitivity analysis was carried out todetermine the sensitivity coefficient for all input parametersand use the sensitivity analysis results to assess the modelsimulation based on the assumption that possibly the cumu-lative effect of input parameters in terms of significant digitsapproximation could be contributing to the underpredictionof the volumetric water content of the simulation In additionsensitivity analysis is one of the most important steps inevaluating the effect of input parameter on simulation resultsand it is also used by others for model validation [13ndash16]
32 Sensitivity Analysis and Simulation Model ValidationNegligible sensitivity response could be due to too small per-turbation size and inaccuracy in sensitivity response couldbe due to too large perturbation size [17] Values of inputparameters were subjected to a perturbation size betweenminus5 and 5 as suggested by Zheng and Bennett [12] andin considering the simulation time we limit the sensitivityanalysis to a simulation time of 105 s The sensitivity analysisstudy was based on a single perturbation size of incrementor decrement in each simulation The sensitivity analysiswas carried out based on the hydraulic functions (10) fromHaverkamp et al [7]
There were seven input parameters from Haverkamphydraulic functions as listed in Table 1 Additional fourinput parameters were also tested that is initial volumetricwater content (120579
119871(initial cond)) boundary volumetric water
content 120579119871(upper cond) time-step size (Δ119905) and spatial
spacing size (Δ119911) The depth at 155 cm from the groundsurface was used for observation
The normalized sensitivity coefficients are shown inFigure 2 Generally there are two groups of sensitivity coef-ficients that is positive and negative relations In positiverelation group the boundary volumetric water content hadthe highest sensitivity coefficientThis was followed by initialvolumetric water content and saturated hydraulic conductiv-ity The smallest sensitivity coefficient in the group was theresidual volumetric water content In negative relation groupsaturated volumetric water content had the highest sensitivitycoefficient and this group endedwith spatial spacing size andtime-step size as the smallest sensitivity coefficient
For comparison purpose elementary effects method wasalso used to calculate normalized sensitivity coefficient Weassumed only random generation in 119896-dimensional diagonalmatrix (Dlowast) and then (2) was used to generate 50 trajectoriesEquation (3) was used to screen out 4 trajectories with thegreatest geometric distance of those trajectories Equations(4) to (7) were used to calculate the elementary effects meanof elementary effects mean of absolute values of the elemen-tary effects and standard deviation respectivelyThemean ofelementary effects was modified to calculate the normalized
100300500700900
1100
Input parameter
minus100
minus300
minus500
minus700
minus900
minus1100
Nor
mal
ized
sens
itivi
ty co
effici
ent (
)
120579L
(upp
er b
ound
)
120579L
(initi
al co
nd)120579s 120579r
Ks120573 A120572Δz
ΔtB
644Eminus01
508E
minus01
208Eminus01
825Eminus02
385Eminus02
minus619Eminus04
minus550E
minus02
minus589Eminus01
minus107E+00
minus497E+00
471E
+00
Figure 2 The rank of sensitivity coefficient Note 120579119904and 120579
119903
are saturated and residual volumetric water content Δ119911 spatialspacing sizeΔ119905 time-step size119870
119904 saturated hydraulic conductivity
120579119871(initial cond) clay medium initial value of volumetric water
content 120579119871(upper bound) upper boundary of volumetric water
content 119860 119861 120573 and 120572 are fitting parameters from Haverkamp asin (10)
Table 2 Statistical measures (120583 120583lowast and 120590) of elementary effectsmethod They are the mean of elementary effects the meanof absolute values of the elementary effects and the standarddeviation respectively Note that 120579
119903is residual volumetric water
content 120579119904is saturated volumetric water content 119870
119904is saturated
hydraulic conductivityΔ119911 is spatial spacing sizeΔ119905 is time-step size120579119871(initial cond) is initial value of volumetric water content and 120572
120573 119860 and 119861 are fitting coefficients
120583 () 120583lowast () 120590
120579119904
minus603119864 + 00 603119864 + 00 948119864 minus 01
119861 minus185119864 + 00 185119864 + 00 938119864 minus 01
120573 minus207119864 minus 01 320119864 minus 01 370119864 minus 01
120572 minus414119864 minus 02 125119864 minus 01 147119864 minus 01
Δ119911 minus335119864 minus 02 395119864 minus 02 456119864 minus 02
Δ119905 minus166119864 minus 04 525119864 minus 04 621119864 minus 04
120579119903
444119864 minus 03 337119864 minus 02 402119864 minus 02
119860 313119864 minus 01 313119864 minus 01 741119864 minus 02
119870119904
524119864 minus 01 524119864 minus 01 673119864 minus 02
120579119871(initial cond) 884119864 minus 01 884119864 minus 01 305119864 minus 01
sensitivity coefficient The results are shown in Table 2 Thesensitivity coefficient has identical ranking as those obtainedin Figure 2 except for the coefficient of 120572 input parameterSimilar values of 120583 and 120583
lowast indicate linear effect on few inputparameters in positive (119860119870
119904 and 120579
119871(initial cond)) and nega-
tive (120579119904and 119861) relations Other input parameters have shown
the effect of oscillating sign that results in different values of120583 and 120583
lowast In general those sensitivity coefficients generatedby different methods have shown comparable results
We assumed that a minor deviation in each input param-eter in terms of its significant digits approximation couldcontribute some effects on the simulation outcome thatcould possibly explain the discrepancy between the simulatedresults and Philiprsquos semianalytical solution (Figure 1) In other
International Scholarly Research Notices 5
Table 3 Significant digits approximation on input parameter valueNote that 120579
119903is residual volumetric water content 120579
119904is saturated
volumetric water content 119870119904is saturated hydraulic conductivity
Δ119911 is spatial spacing size Δ119905 is time-step size 120579119871(initial cond) is
initial value of volumetric water content 120579119871(upper bond) is upper
boundary of volumetric water content and 120572 120573119860 and 119861 are fittingcoefficients
Δ119905 10 s the base case was 500 sΔ119911 01 cm the base case was 1 cm
words the parameter values in terms of significant digitsapproximation that were used in computer simulation byHaverkamp et al [7] could be different from the exactdata in terms of input parameter significant digits thatthey published Thus we take advantage on the positive andnegative relations generated from the sensitivity analysis andset up a hypothetical approximation value in Table 3 forfurther investigation The cumulative effect was studied bymanipulating an input parameter used for each simulationand the subsequent manipulation of input parameter wascarried out on top of the previous changed input parameterThis process begins from step 1 for base case to step 10 forspatial spacing size For instance the 120579
119871(initial cond) value
(02376499m3mminus3) was used as a second simulation (in step2) after the base case simulation This was followed by thirdsimulation (in step 3) using 120579
119903value as 0124499m3mminus3
by remaining 120579119871(initial cond) value used in the second
simulation For each simulation Equation (11) was used tocalculate the discrepancy between simulation results andPhiliprsquos semianalytical solution (data from [7]) for absoluteresidual error (MA)Of all those eleven parameters inTable 3Δ119905 and Δ119911 were the only two parameters without any limit ofvariation and for this reason we extend the variation limitby reducing the former and the latter by 98 and 90 from500 s and 1 cm to 10 s and 01 cm respectively The 120579
119904and
120579119871(119906119901119901119890119903 119887119900119906119899119889) values are negative and positive relations
respectively Decreasing and increasing the correspondingformer and latter values would result in simulation failurethus those two parameters remained unchanged
A consistent reduction inMA value from 120579119871(initial cond)
to 119861 input parameter was observed except a slight incrementat Δ119905 input parameter simulation and a steep slide of MAvalue was observed on the Δ119911 input parameter simulation(Figure 3) Although the sensitivity coefficient in Figure 2indicates that reducing Δ119905 value should lead to a reductionin MA value the simulated result showed an increase in
002
54
002
53
002
53
002
45
002
45
002
45
002
45
002
338
002
343
001
06
0010
0015
0020
0025
0030
0035
Input parameter
Abso
lute
resid
ual e
rror
(MA
)
Δz
(ste
p10
)
Δt
(ste
p9
)
B(s
tep8
)
A(s
tep7
)
Ks
(ste
p6
)
120572(s
tep5
)
120573(s
tep4
)
120579L
(initi
al co
nd)
(ste
p2
)
Base
case
(ste
p 1)
120579r
(ste
p3
)
(m3
mminus3)
Figure 3The cumulative effect of input parameters on the absoluteresidual error at simulation time 105 s Note 120579
119871(initial cond) (step
2) claymedium initial value of volumetric water content 120579119903(step 3)
residual volumetric water content 120573 (step 4) 120572 (step 5) 119860 (step 7)and 119861 (step 8) are fitting parameters119870
the MA value This observation could be explained from theresult of elementary effects method This was because Δ119905 hasdifferent values of 120583 and 120583
lowast which indicate the capability ofsign oscillation (Table 2)
Figure 3 shows that the simulation on the cumulativeeffect of steps 2ndash9 which combined the effect from 120579
119871(initial
cond) (step 2) with Δ119905 (step 9) did not contribute to anysignificant effects on the advancement of water infiltrationfront It only resulted in a reduction of 78 in MA valuefrom 00254 to 002343m3mminus3 In addition those eight inputparameters had to vary in significant digits approximation astabulated in Table 3 in order to result in the stated percentagereduction Therefore the significant digits approximationmight not be the main cause of the problem in consideringthat a more significant effect on the advance of water infiltra-tion front was shown by Δ119911 in the Figure 4 A further step toinclude Δ119911 in the simulation that is the cumulative effect ofsteps 2ndash10 which combined the effect from 120579
119871(initial cond)
(step 2) with Δ119911 (step 10) there was 547 reduction in MAvalue of step 9 from 002343 to 00106m3mminus3 This indicatesthat the spatial spacing size was themain cause in the advanceof water infiltration front Therefore the simulation wasrepeated for the last time for the effect of spatial spacing sizealone and in Figure 4 there was a good agreement betweenthe simulation results and the Philip(K) This observationcould be explained using (1) after rearranging it into thefollowing form which we termed as percentage variation insimulation results
Δ119910119894
119910119894
=Δ119886119908
119886119908
119883119894119908 (12)
6 International Scholarly Research Notices
16
17
18
19
20
21
22
23
027 029 031 033 035 037 039
Philip(H)
Base case Philip(K)Only step 10
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Steps 2ndash10Steps 2ndash9
Figure 4 The effect of Δ119911 (step 10 alone) and cumulative effects ofsteps 2 to 9 and 2ndash10 in comparison with Philip(H) and Philip(K)
where Δ119886119908119886119908
is the normalized input parameter valueΔ119910119894119910119894is the normalized output parameter value and 119883
119894119908
is the normalized sensitivity coefficient () Equation (12)is simply a multiplication of the percentage change in inputparameter value from the base case and the normalizedsensitivity coefficient
Using (12) the percentage variation in simulation resultsfrom input parameters of Δ119911 and Δ119905 caused an incrementof 495 and 006 respectively despite Δ119905 having thehighest reduction in percentage (minus98) from base case Thisobservation could be summarized as follows firstly inputparameter with the highest sensitivity coefficient does notguarantee the greatest effect on the simulation result forexample 120579
119871(initial cond) secondly input parameter with
the highest percentage of change also does not guaranteethe greatest effect on the simulation result for exampleΔ119905 and therefore only the highest sensitivity coefficientwith the highest percentage change on input parameter (orthe uncertainty) would give the most substantial effect onsimulation result
4 Conclusions
Thegoverning equation of transientwater flow in unsaturatedand nonisothermal conditions was approximated numeri-cally by finite-difference solution It was successfully imple-mented into FORTRAN programming language simulatedand verified by Philiprsquos semianalytical solution on waterinfiltration into Yolo light clay with data from literatures
One-at-a-timeOAT and elementary effects EEmethodswere used in the sensitivity analysis A common trend ofsensitivity was observed across the methods in both positiveand negative relationsThe latter method allowed explorationof additional characteristics of input parameters at differentinput space such as linearity and sign oscillation effectThe sign oscillation effect observed on input parametersexplained the possibility of its deviation from those observedin OAT method at different input spaces
A hypothetical case that was established to study thecumulative effect of input parameters on the discrepancybetween simulated result and Philiprsquos semianalytical solutionin terms of significant digits approximation (from base case)was found to be unlikely A large normalized sensitivitycoefficient was made with initial volumetric water contentand the largest percentage changes were with time-step sizebut surprisingly none of them contributes to any substantialimpact on simulation results when compared to spatialspacing size This observation led to the conclusion that theuncertainty of input parameter and normalized sensitivitycoefficient of input parameters both controlled the outcomeof simulation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportfrom Meiji University Japan and the Ministry of EducationMalaysia and alsoUniversitiMalaysia TerengganuMalaysia
References
[1] D J Pannell ldquoSensitivity analysis of normative economic mod-els theoretical framework and practical strategiesrdquoAgriculturalEconomics vol 16 no 2 pp 139ndash152 1997
[2] D M Hamby ldquoA review of techniques for parameter sensitivityanalysis of environmental modelsrdquo Environmental Monitoringand Assessment vol 32 no 2 pp 135ndash154 1994
[3] A Saltelli S Tarantola F Campolongo andMRatto SensitivityAnalysis in Practice A Guide to Assessing Scientific Models JohnWiley amp Sons Hoboken NJ USA 2004
[4] A Saltelli and P Annoni ldquoHow to avoid a perfunctory sensitiv-ity analysisrdquo Environmental Modelling and Software vol 25 no12 pp 1508ndash1517 2010
[5] A Saltelli M Ratto and T Andres Global Sensitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008
[6] F Campolongo J Cariboni and A Saltelli ldquoAn effectivescreening design for sensitivity analysis of large modelsrdquo Envi-ronmental Modelling and Software vol 22 no 10 pp 1509ndash15182007
[7] R Haverkamp M Vauclin J Touma P J Wierenga and GVachaud ldquoA comparison of numerical simulation models forone-dimensional infiltrationrdquo Soil Science Society of AmericaJournal vol 41 no 2 pp 285ndash294 1977
[8] Z J Kabala and P C D Milly ldquoSensitivity analysis of flowin unsaturated heterogeneous porous media theory numericalmodel and its verificationrdquo Water Resources Research vol 26no 4 pp 593ndash610 1990
[9] M A Celia E T Bouloutas and R L Zarba ldquoA generalmass-conservative numerical solution for the unsaturated flowequationrdquo Water Resources Research vol 26 no 7 pp 1483ndash1496 1990
[10] J Tu G H Yeoh and C Liu Computational Fluid DynamicsA Practical Approach Butterworth-Heinemann Oxford UK2008
International Scholarly Research Notices 7
[11] J Istok ldquoStep 4 solve system of equationsrdquo in GroundwaterModeling by the Finite Element Method chapter 5 pp 176ndash225American Geophysical Union Washington DC USA 1989
[12] C Zheng and G D Bennett Applied Contaminant TransportModeling JohnWiley amp Sons New York NY USA 2nd edition2002
[13] F Stange K Butterbach-Bahl H Papen S Zechmeister-Boltenstern C Li and J Aber ldquoA process-oriented model ofN2O and NO emissions from forest soils 2 Sensitivity analysis
and validationrdquo Journal of Geophysical Research Atmospheresvol 105 no 4 Article ID 1999JD900948 pp 4385ndash4398 2000
[14] R Nathan U N Safriel and I Noy-Meir ldquoField validation andsensitivity analysis of a mechanistic model for tree seed dis-persal by windrdquo Ecology vol 82 no 2 pp 374ndash388 2001
[15] C HMin Y L He X L Liu B H YinW Jiang andW Q TaoldquoParameter sensitivity examination and discussion of PEM fuelcell simulation model validation Part II Results of sensitivityanalysis and validation of the modelrdquo Journal of Power Sourcesvol 160 no 1 pp 374ndash385 2006
[16] S N Gosling and N W Arnell ldquoSimulating current globalriver runoff with a global hydrological model model revisionsvalidation and sensitivity analysisrdquo Hydrological Processes vol25 no 7 pp 1129ndash1145 2011
[17] E P Poeter and M C Hill Documentation of UCODE A Com-puter Code for Universal Inverse Modeling vol 98 DIANEPublishing 1998
Δ119905 10 s the base case was 500 sΔ119911 01 cm the base case was 1 cm
words the parameter values in terms of significant digitsapproximation that were used in computer simulation byHaverkamp et al [7] could be different from the exactdata in terms of input parameter significant digits thatthey published Thus we take advantage on the positive andnegative relations generated from the sensitivity analysis andset up a hypothetical approximation value in Table 3 forfurther investigation The cumulative effect was studied bymanipulating an input parameter used for each simulationand the subsequent manipulation of input parameter wascarried out on top of the previous changed input parameterThis process begins from step 1 for base case to step 10 forspatial spacing size For instance the 120579
119871(initial cond) value
(02376499m3mminus3) was used as a second simulation (in step2) after the base case simulation This was followed by thirdsimulation (in step 3) using 120579
119903value as 0124499m3mminus3
by remaining 120579119871(initial cond) value used in the second
simulation For each simulation Equation (11) was used tocalculate the discrepancy between simulation results andPhiliprsquos semianalytical solution (data from [7]) for absoluteresidual error (MA)Of all those eleven parameters inTable 3Δ119905 and Δ119911 were the only two parameters without any limit ofvariation and for this reason we extend the variation limitby reducing the former and the latter by 98 and 90 from500 s and 1 cm to 10 s and 01 cm respectively The 120579
119904and
120579119871(119906119901119901119890119903 119887119900119906119899119889) values are negative and positive relations
respectively Decreasing and increasing the correspondingformer and latter values would result in simulation failurethus those two parameters remained unchanged
A consistent reduction inMA value from 120579119871(initial cond)
to 119861 input parameter was observed except a slight incrementat Δ119905 input parameter simulation and a steep slide of MAvalue was observed on the Δ119911 input parameter simulation(Figure 3) Although the sensitivity coefficient in Figure 2indicates that reducing Δ119905 value should lead to a reductionin MA value the simulated result showed an increase in
002
54
002
53
002
53
002
45
002
45
002
45
002
45
002
338
002
343
001
06
0010
0015
0020
0025
0030
0035
Input parameter
Abso
lute
resid
ual e
rror
(MA
)
Δz
(ste
p10
)
Δt
(ste
p9
)
B(s
tep8
)
A(s
tep7
)
Ks
(ste
p6
)
120572(s
tep5
)
120573(s
tep4
)
120579L
(initi
al co
nd)
(ste
p2
)
Base
case
(ste
p 1)
120579r
(ste
p3
)
(m3
mminus3)
Figure 3The cumulative effect of input parameters on the absoluteresidual error at simulation time 105 s Note 120579
119871(initial cond) (step
2) claymedium initial value of volumetric water content 120579119903(step 3)
residual volumetric water content 120573 (step 4) 120572 (step 5) 119860 (step 7)and 119861 (step 8) are fitting parameters119870
the MA value This observation could be explained from theresult of elementary effects method This was because Δ119905 hasdifferent values of 120583 and 120583
lowast which indicate the capability ofsign oscillation (Table 2)
Figure 3 shows that the simulation on the cumulativeeffect of steps 2ndash9 which combined the effect from 120579
119871(initial
cond) (step 2) with Δ119905 (step 9) did not contribute to anysignificant effects on the advancement of water infiltrationfront It only resulted in a reduction of 78 in MA valuefrom 00254 to 002343m3mminus3 In addition those eight inputparameters had to vary in significant digits approximation astabulated in Table 3 in order to result in the stated percentagereduction Therefore the significant digits approximationmight not be the main cause of the problem in consideringthat a more significant effect on the advance of water infiltra-tion front was shown by Δ119911 in the Figure 4 A further step toinclude Δ119911 in the simulation that is the cumulative effect ofsteps 2ndash10 which combined the effect from 120579
119871(initial cond)
(step 2) with Δ119911 (step 10) there was 547 reduction in MAvalue of step 9 from 002343 to 00106m3mminus3 This indicatesthat the spatial spacing size was themain cause in the advanceof water infiltration front Therefore the simulation wasrepeated for the last time for the effect of spatial spacing sizealone and in Figure 4 there was a good agreement betweenthe simulation results and the Philip(K) This observationcould be explained using (1) after rearranging it into thefollowing form which we termed as percentage variation insimulation results
Δ119910119894
119910119894
=Δ119886119908
119886119908
119883119894119908 (12)
6 International Scholarly Research Notices
16
17
18
19
20
21
22
23
027 029 031 033 035 037 039
Philip(H)
Base case Philip(K)Only step 10
Dep
thz
(cm
)
Volumetric water content (m3 mminus3)
Steps 2ndash10Steps 2ndash9
Figure 4 The effect of Δ119911 (step 10 alone) and cumulative effects ofsteps 2 to 9 and 2ndash10 in comparison with Philip(H) and Philip(K)
where Δ119886119908119886119908
is the normalized input parameter valueΔ119910119894119910119894is the normalized output parameter value and 119883
119894119908
is the normalized sensitivity coefficient () Equation (12)is simply a multiplication of the percentage change in inputparameter value from the base case and the normalizedsensitivity coefficient
Using (12) the percentage variation in simulation resultsfrom input parameters of Δ119911 and Δ119905 caused an incrementof 495 and 006 respectively despite Δ119905 having thehighest reduction in percentage (minus98) from base case Thisobservation could be summarized as follows firstly inputparameter with the highest sensitivity coefficient does notguarantee the greatest effect on the simulation result forexample 120579
119871(initial cond) secondly input parameter with
the highest percentage of change also does not guaranteethe greatest effect on the simulation result for exampleΔ119905 and therefore only the highest sensitivity coefficientwith the highest percentage change on input parameter (orthe uncertainty) would give the most substantial effect onsimulation result
4 Conclusions
Thegoverning equation of transientwater flow in unsaturatedand nonisothermal conditions was approximated numeri-cally by finite-difference solution It was successfully imple-mented into FORTRAN programming language simulatedand verified by Philiprsquos semianalytical solution on waterinfiltration into Yolo light clay with data from literatures
One-at-a-timeOAT and elementary effects EEmethodswere used in the sensitivity analysis A common trend ofsensitivity was observed across the methods in both positiveand negative relationsThe latter method allowed explorationof additional characteristics of input parameters at differentinput space such as linearity and sign oscillation effectThe sign oscillation effect observed on input parametersexplained the possibility of its deviation from those observedin OAT method at different input spaces
A hypothetical case that was established to study thecumulative effect of input parameters on the discrepancybetween simulated result and Philiprsquos semianalytical solutionin terms of significant digits approximation (from base case)was found to be unlikely A large normalized sensitivitycoefficient was made with initial volumetric water contentand the largest percentage changes were with time-step sizebut surprisingly none of them contributes to any substantialimpact on simulation results when compared to spatialspacing size This observation led to the conclusion that theuncertainty of input parameter and normalized sensitivitycoefficient of input parameters both controlled the outcomeof simulation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportfrom Meiji University Japan and the Ministry of EducationMalaysia and alsoUniversitiMalaysia TerengganuMalaysia
References
[1] D J Pannell ldquoSensitivity analysis of normative economic mod-els theoretical framework and practical strategiesrdquoAgriculturalEconomics vol 16 no 2 pp 139ndash152 1997
[2] D M Hamby ldquoA review of techniques for parameter sensitivityanalysis of environmental modelsrdquo Environmental Monitoringand Assessment vol 32 no 2 pp 135ndash154 1994
[3] A Saltelli S Tarantola F Campolongo andMRatto SensitivityAnalysis in Practice A Guide to Assessing Scientific Models JohnWiley amp Sons Hoboken NJ USA 2004
[4] A Saltelli and P Annoni ldquoHow to avoid a perfunctory sensitiv-ity analysisrdquo Environmental Modelling and Software vol 25 no12 pp 1508ndash1517 2010
[5] A Saltelli M Ratto and T Andres Global Sensitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008
[6] F Campolongo J Cariboni and A Saltelli ldquoAn effectivescreening design for sensitivity analysis of large modelsrdquo Envi-ronmental Modelling and Software vol 22 no 10 pp 1509ndash15182007
[7] R Haverkamp M Vauclin J Touma P J Wierenga and GVachaud ldquoA comparison of numerical simulation models forone-dimensional infiltrationrdquo Soil Science Society of AmericaJournal vol 41 no 2 pp 285ndash294 1977
[8] Z J Kabala and P C D Milly ldquoSensitivity analysis of flowin unsaturated heterogeneous porous media theory numericalmodel and its verificationrdquo Water Resources Research vol 26no 4 pp 593ndash610 1990
[9] M A Celia E T Bouloutas and R L Zarba ldquoA generalmass-conservative numerical solution for the unsaturated flowequationrdquo Water Resources Research vol 26 no 7 pp 1483ndash1496 1990
[10] J Tu G H Yeoh and C Liu Computational Fluid DynamicsA Practical Approach Butterworth-Heinemann Oxford UK2008
International Scholarly Research Notices 7
[11] J Istok ldquoStep 4 solve system of equationsrdquo in GroundwaterModeling by the Finite Element Method chapter 5 pp 176ndash225American Geophysical Union Washington DC USA 1989
[12] C Zheng and G D Bennett Applied Contaminant TransportModeling JohnWiley amp Sons New York NY USA 2nd edition2002
[13] F Stange K Butterbach-Bahl H Papen S Zechmeister-Boltenstern C Li and J Aber ldquoA process-oriented model ofN2O and NO emissions from forest soils 2 Sensitivity analysis
and validationrdquo Journal of Geophysical Research Atmospheresvol 105 no 4 Article ID 1999JD900948 pp 4385ndash4398 2000
[14] R Nathan U N Safriel and I Noy-Meir ldquoField validation andsensitivity analysis of a mechanistic model for tree seed dis-persal by windrdquo Ecology vol 82 no 2 pp 374ndash388 2001
[15] C HMin Y L He X L Liu B H YinW Jiang andW Q TaoldquoParameter sensitivity examination and discussion of PEM fuelcell simulation model validation Part II Results of sensitivityanalysis and validation of the modelrdquo Journal of Power Sourcesvol 160 no 1 pp 374ndash385 2006
[16] S N Gosling and N W Arnell ldquoSimulating current globalriver runoff with a global hydrological model model revisionsvalidation and sensitivity analysisrdquo Hydrological Processes vol25 no 7 pp 1129ndash1145 2011
[17] E P Poeter and M C Hill Documentation of UCODE A Com-puter Code for Universal Inverse Modeling vol 98 DIANEPublishing 1998
Figure 4 The effect of Δ119911 (step 10 alone) and cumulative effects ofsteps 2 to 9 and 2ndash10 in comparison with Philip(H) and Philip(K)
where Δ119886119908119886119908
is the normalized input parameter valueΔ119910119894119910119894is the normalized output parameter value and 119883
119894119908
is the normalized sensitivity coefficient () Equation (12)is simply a multiplication of the percentage change in inputparameter value from the base case and the normalizedsensitivity coefficient
Using (12) the percentage variation in simulation resultsfrom input parameters of Δ119911 and Δ119905 caused an incrementof 495 and 006 respectively despite Δ119905 having thehighest reduction in percentage (minus98) from base case Thisobservation could be summarized as follows firstly inputparameter with the highest sensitivity coefficient does notguarantee the greatest effect on the simulation result forexample 120579
119871(initial cond) secondly input parameter with
the highest percentage of change also does not guaranteethe greatest effect on the simulation result for exampleΔ119905 and therefore only the highest sensitivity coefficientwith the highest percentage change on input parameter (orthe uncertainty) would give the most substantial effect onsimulation result
4 Conclusions
Thegoverning equation of transientwater flow in unsaturatedand nonisothermal conditions was approximated numeri-cally by finite-difference solution It was successfully imple-mented into FORTRAN programming language simulatedand verified by Philiprsquos semianalytical solution on waterinfiltration into Yolo light clay with data from literatures
One-at-a-timeOAT and elementary effects EEmethodswere used in the sensitivity analysis A common trend ofsensitivity was observed across the methods in both positiveand negative relationsThe latter method allowed explorationof additional characteristics of input parameters at differentinput space such as linearity and sign oscillation effectThe sign oscillation effect observed on input parametersexplained the possibility of its deviation from those observedin OAT method at different input spaces
A hypothetical case that was established to study thecumulative effect of input parameters on the discrepancybetween simulated result and Philiprsquos semianalytical solutionin terms of significant digits approximation (from base case)was found to be unlikely A large normalized sensitivitycoefficient was made with initial volumetric water contentand the largest percentage changes were with time-step sizebut surprisingly none of them contributes to any substantialimpact on simulation results when compared to spatialspacing size This observation led to the conclusion that theuncertainty of input parameter and normalized sensitivitycoefficient of input parameters both controlled the outcomeof simulation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportfrom Meiji University Japan and the Ministry of EducationMalaysia and alsoUniversitiMalaysia TerengganuMalaysia
References
[1] D J Pannell ldquoSensitivity analysis of normative economic mod-els theoretical framework and practical strategiesrdquoAgriculturalEconomics vol 16 no 2 pp 139ndash152 1997
[2] D M Hamby ldquoA review of techniques for parameter sensitivityanalysis of environmental modelsrdquo Environmental Monitoringand Assessment vol 32 no 2 pp 135ndash154 1994
[3] A Saltelli S Tarantola F Campolongo andMRatto SensitivityAnalysis in Practice A Guide to Assessing Scientific Models JohnWiley amp Sons Hoboken NJ USA 2004
[4] A Saltelli and P Annoni ldquoHow to avoid a perfunctory sensitiv-ity analysisrdquo Environmental Modelling and Software vol 25 no12 pp 1508ndash1517 2010
[5] A Saltelli M Ratto and T Andres Global Sensitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008
[6] F Campolongo J Cariboni and A Saltelli ldquoAn effectivescreening design for sensitivity analysis of large modelsrdquo Envi-ronmental Modelling and Software vol 22 no 10 pp 1509ndash15182007
[7] R Haverkamp M Vauclin J Touma P J Wierenga and GVachaud ldquoA comparison of numerical simulation models forone-dimensional infiltrationrdquo Soil Science Society of AmericaJournal vol 41 no 2 pp 285ndash294 1977
[8] Z J Kabala and P C D Milly ldquoSensitivity analysis of flowin unsaturated heterogeneous porous media theory numericalmodel and its verificationrdquo Water Resources Research vol 26no 4 pp 593ndash610 1990
[9] M A Celia E T Bouloutas and R L Zarba ldquoA generalmass-conservative numerical solution for the unsaturated flowequationrdquo Water Resources Research vol 26 no 7 pp 1483ndash1496 1990
[10] J Tu G H Yeoh and C Liu Computational Fluid DynamicsA Practical Approach Butterworth-Heinemann Oxford UK2008
International Scholarly Research Notices 7
[11] J Istok ldquoStep 4 solve system of equationsrdquo in GroundwaterModeling by the Finite Element Method chapter 5 pp 176ndash225American Geophysical Union Washington DC USA 1989
[12] C Zheng and G D Bennett Applied Contaminant TransportModeling JohnWiley amp Sons New York NY USA 2nd edition2002
[13] F Stange K Butterbach-Bahl H Papen S Zechmeister-Boltenstern C Li and J Aber ldquoA process-oriented model ofN2O and NO emissions from forest soils 2 Sensitivity analysis
and validationrdquo Journal of Geophysical Research Atmospheresvol 105 no 4 Article ID 1999JD900948 pp 4385ndash4398 2000
[14] R Nathan U N Safriel and I Noy-Meir ldquoField validation andsensitivity analysis of a mechanistic model for tree seed dis-persal by windrdquo Ecology vol 82 no 2 pp 374ndash388 2001
[15] C HMin Y L He X L Liu B H YinW Jiang andW Q TaoldquoParameter sensitivity examination and discussion of PEM fuelcell simulation model validation Part II Results of sensitivityanalysis and validation of the modelrdquo Journal of Power Sourcesvol 160 no 1 pp 374ndash385 2006
[16] S N Gosling and N W Arnell ldquoSimulating current globalriver runoff with a global hydrological model model revisionsvalidation and sensitivity analysisrdquo Hydrological Processes vol25 no 7 pp 1129ndash1145 2011
[17] E P Poeter and M C Hill Documentation of UCODE A Com-puter Code for Universal Inverse Modeling vol 98 DIANEPublishing 1998
[11] J Istok ldquoStep 4 solve system of equationsrdquo in GroundwaterModeling by the Finite Element Method chapter 5 pp 176ndash225American Geophysical Union Washington DC USA 1989
[12] C Zheng and G D Bennett Applied Contaminant TransportModeling JohnWiley amp Sons New York NY USA 2nd edition2002
[13] F Stange K Butterbach-Bahl H Papen S Zechmeister-Boltenstern C Li and J Aber ldquoA process-oriented model ofN2O and NO emissions from forest soils 2 Sensitivity analysis
and validationrdquo Journal of Geophysical Research Atmospheresvol 105 no 4 Article ID 1999JD900948 pp 4385ndash4398 2000
[14] R Nathan U N Safriel and I Noy-Meir ldquoField validation andsensitivity analysis of a mechanistic model for tree seed dis-persal by windrdquo Ecology vol 82 no 2 pp 374ndash388 2001
[15] C HMin Y L He X L Liu B H YinW Jiang andW Q TaoldquoParameter sensitivity examination and discussion of PEM fuelcell simulation model validation Part II Results of sensitivityanalysis and validation of the modelrdquo Journal of Power Sourcesvol 160 no 1 pp 374ndash385 2006
[16] S N Gosling and N W Arnell ldquoSimulating current globalriver runoff with a global hydrological model model revisionsvalidation and sensitivity analysisrdquo Hydrological Processes vol25 no 7 pp 1129ndash1145 2011
[17] E P Poeter and M C Hill Documentation of UCODE A Com-puter Code for Universal Inverse Modeling vol 98 DIANEPublishing 1998
![arXiv:1705.03260v1 [cs.AI] 9 May 2017 · 2018. 10. 14. · Vegetables2 Normalized Log Size Vehicles1 Normalized Log Size Vehicles2 Normalized Log Size Weapons1 Normalized Log Size](https://static.documents.pub/doc/80x56/5ff2638300ded74c7a39596f/arxiv170503260v1-csai-9-may-2017-2018-10-14-vegetables2-normalized-log.jpg)
