Comparative analysis of intelligent models for prediction of Langmuir constants for CO 2 adsorption

13 Pages·2016·1.79 MB·English

by  A. K. Verma

Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109
DOI10.1007/s40948-016-0025-3
ORIGINAL ARTICLE
Comparative analysis of intelligent models for prediction
of Langmuir constants for CO adsorption of Gondwana
2
coals in India
.
A. K. Verma Abhinav Sirvaiya
Received:20October2015/Accepted:22March2016/Publishedonline:29March2016
(cid:2)SpringerInternationalPublishingSwitzerland2016
Abstract An artificial neural network (ANN) is an Keywords ANN(cid:2)Sequestration(cid:2)CO (cid:2)Gondwana
2
artificial intelligence technique in which performance coals(cid:2)Langmuirvolumeconstant(cid:2)Langmuirpressure
can be improved by adapting to the changes in the constant
environment. The efficient manipulation of large
amounts of data and the ability to generalize results
arethemainadvantagesofneuralnetworks.Consider-
ingtheadvantagesofthistechnique,thispresentpaper 1 Introduction
aimstoperformacomparisonbetweenlinearmethods
like Multivariate Regression Analysis (MVRA) and Coalbedmethane(CBM)productionwith CO injec-
2
different ANN techniques such as back propagation tion combination is presently a worldwide topic of
withregressionanalysis(BPNN),layerrecurrentneural researchandstudies.TheroleofCO injectionisnot
2
network (LRNN), generalized regression neural net- onlytoenhanceCBMproduction(ECBM)butalsoin
work(GRNN)andradialbasisneuralnetwork(RBNN). subsurfacestorage ofhugeamountofCO which are
2
Thiscomparisonwasperformedtopredicttheapprox- contributing in reduction of greenhouse gas percent-
imatevaluesofLangmuirvolumeconstant(LVC)and age in atmosphere. Storing CO in deep unminable
2
Langmuirpressureconstant(LPC)forCO adsorption coalbedshasbecomeoneofthepromisingtechnolo-
2
in coal using proximate and maceral properties of gies of geological sequestration (Busch and Gen-
India’smajorcoalfieldasinputparameters.Itisfound sterblum2011).
that RMSE value for RBNN is least followed by TheentiremethanegasinaCoalbedMethane(CBM)
GRNN,LRNN,BPNNandMVRAforbothLVCand reservoirisstoredintothemicroporesandcleatswithin
LPCmodels.Basedonthebestnetwork,itisfoundthat thecoalmatrixinadsorbedform.Forenhancepurpose,
coal seams from Narayankuri coal mine has highest theCO capturedisbeinginjectedintothesubsurface
2
adsorbing capacity of CO (0.0019791 mol/gm) as coalbedswhich getadsorbtothecoalporessurfaces.
2
comparedtoothercoalseamsofthisstudy. CO canmoveinthefinestporesandadsorbtothecoal
2
atanear-liquiddensityfirmlywithnominalchancesof
being released lately. (Krooss et al. 2002). Coal has
higheraffinitytoCO gas.Thus,CO replacesadsorbed
2 2
A.K.Verma(&)(cid:2)A.Sirvaiya methanefromthecoal,acceleratingmethanerecovery
DepartmentofMiningEngineering,IndianSchoolof
inECBMprocess.Therefore,CO forECBMisavalue-
Mines,Dhanbad826004,Jharkhand,India 2
addedselection.Although,relativesorptionaffinityof
e-mail:[email protected];
[email protected] coaltomethaneandCO2istheprimaryparameterfactor
123
98 Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109
which is needed to be consider for site suitability parameters help in determining of CO adsorption
2
analysis of CO –ECBM (Dutta et al.,2011). A good tendency in coals. Generally it has been seen that the
2
understandingofmethaneandCO sorptionbehavioron coal with higher Langmuir pressure constant value
2
coalisanunderlyingrequirementfortheestimationof shows very high sorption capacity in low pressure
methanerecoveryandCO storagecapacityofacoal. range.Itwasobservedthattheadsorptionisothermsof
2
In coal matrix, mainly the CO storage occurs by the coals if are steadily increasing and remain under-
2
physical sorption (e.g. Harpalani and Chen 1997). saturated within the experimental pressure range then
Therearemanyfactorsthatareneedtobeconsidered thesecoalshavecomparableorevenhigherLangmuir
forunderstandingthecomplexityoftheinteractionof volumeconstant.
CO withcoalinthecleatsystemandthecoalmatrix. Thispaperhasalinearandnonlinearmathematical
2
Different aspects like moisture, volatile matter per- modelslikeMultivariateregressionanalysis(MVRA),
centage, Ash content, depth of coal formation, ArtificialNeuralNetwork(ANN),RadialBiasneural
Liptinite, Inertinite, Mineral Matter etc. that are network(RBNN),andLayerRecurrentmethodwhich
obtained from the laboratory are minimum require- are used to calculate Langmuir constants (for CO )
2
menttodealwithCO adsorptionselectivity. using coal proximate and macerals properties like
2
During CO -ECBM processes, the injected CO moisture, volatile matter, ash content, fixed carbon
2 2
flows through network of cleats by a complex mech- composition, vitrinite, semi-vitrinite, liptinite, iner-
anism of sorption/diffusion process (Mazzotti et al. tinite, mineral matter, mean and depth (Dutta et al.
2009). It gets adsorb to the coal inner surface by 2011).Theseparametersaretakenasinputparameters
replacingCH4fromthewalls.Thisreplacementoccurs forthesemathematicalmodels.Themainobjectiveof
dueto(1)byareductionoftheCH4partialpressureor thisstudyistodevelopintelligentmodelstocalculate
(2) by a higher selective sorption of CO over CH4. coalhavingbestadsorptioncapacitytowardsCO gas
2 2
Concentration gradient between CH4 in the matrix forECBMprocess.
compared to the cleat system, results CH4 to diffuse
from the coal matrix into the cleat system where, by
pressure drawdown towards a production well, it is
produced. 2 Studyareaanddataset
Manymethodshavebeenputuptomodelsorption
isotherms on coal and Langmuir isotherm is among A previous study is the source of this study data set
such simple methods. Langmuir isotherm is used to (Dutta et al. 2011). Fourteen coal samples were
quantifytheamountofgasadsorptiononanadsorbent taken from the India’s major coal mining and CMB
as a function of partial pressure or concentration at a exploration activity regions. Eight samples are from
certain temperature. The Langmuir volume constant the Raniganj coalfield, four samples are from the
and Langmuir pressure constant are the two constant Jharia coalfield, and the remaining two are from the
that define the Langmuir isotherm model. It is well South Karanpura Coalfield (Fig. 2). Dry samples are
known approach in the ECBM industry and related the source for analysis. The residual moisture of
reservoir simulations. It provides a reasonable fit to samples is being represent by the moisture content
mostexperimentaldata: of a coal sample. The ash content is high in the
samples which strongly reduces the adsorption of a
V (cid:3)p
n ¼ L coal for CO gas. It is found out that the Raniganj
ads P þp 2
L formation coals have higher moisture content and
Here V is the Langmuir volume, representing the lower ash content than other samples. 32–45 % of
L
amountofgassorbedatinfinitepressureandP isthe Volatile matter content has been found out in the
L
Langmuirpressure(Fig. 1),equivalenttothepressure coal samples. Other parameters like carbon per-
at which half of the Langmuir volume V is reached centage, Liptinite and Inertinite which are helping
L
(Crosdaleetal.1998a,b). in knowing organic percentage in coal etc. have
TheadsorptionofCO incoalbedscanbeexplained also been obtained from experiments to understand
2
by the variation in Langmuir volume and Langmuir sorption capacity of coal for CO gas (Verma and
2
pressure constants (for CO ). These two constant Sirvaiya 2015) (Table 1).
2
123
Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109 99
Fig.1 Typicallangmuirvolumeandpressurecurve
Fig.2 Locationofcoal
samplesfromcoalfieldsin
India(Duttaetal.2011)
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100 Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109
Langmuirpressureconstant.MPa)(PL 2.732 2.406 2.586 1.976 1.074 0.8 1.271 1.167 0.841 0.984 1.082 1.418 1.231 1.161
muirmeantmL/g)
Langvoluconst,(VL 74 68 89 58 25 18 31 26 33 41 35 42 42 39
Depth(m)(d) 160 107 46 190 172 118 201 109 914 996 520 650 450 380
an% 4 2 4 1 6 1 6 6 4 9 2 1 7
Mero(w) 0.6 0.6 0.6 0.6 0.9 1.0 0.9 1.0 1.9 1.3 1.2 1.2 1.1 0.9
Mineralmatter(%)(z) 5.7 7.4 7.9 11 18.1 17.4 8.1 16.9 36.5 81.2 8 6.8 5.3 4.5
e
Inertinit(%)(i) 15.8 16.6 11.4 27.2 61.3 62.1 64.6 61.9 53.4 18.6 5.5 29.5 33.7 9.8
e
Liptinit(%)(l) 4.2 9.7 6 11.3 7.3 5.8 8.3 12.8 0.3 0 0.2 0.3 0.3 1
study Semi-vitrinite(%)(s) 0.4 1 0.8 0.8 1.3 1.8 2.3 0.7 0.5 0 0.2 1.4 2.3 0.5
e
h
plesfort Vitrinite(%)(k) 73.9 65.3 73.8 49.7 12 2.9 16.7 7.7 9.3 0.2 86.1 62 58.4 84.2
m
sa don(f)
coal Fixecarb(%) 45 44 37 39 34.8 34.2 33.9 32.8 33.6 18.2 33.8 37.9 25 39.5
of
ants Ash(%)(a) 10 12 15 17 28 22 30 27 31 48 20 24 34 22
st
n
gmuirco Volatilematter(%)(v) 38 40 41 40 36 41.5 35 38 34 32 45 36.5 35 38
n
a
ertiesandl Moisture(%)(m) 7 4 7 4 1.2 2.3 1.1 2.2 1.4 1.8 1.2 1.6 6 0.5
hysicalprop Nameofthecoalmines Bogra Kenda Narayankuri Satgram Kalimati LocalII Mehaladih Mugmaspecial SKAC1 SKAC3 15thSeam 16thTopseam 16thBottomseam 18thSeam
p
o
Table1Petr Locations RaniganjformationofRaniganjcoalfield BarakarformationofRaniganjcoalfield SouthKaranpuracoalfield Jhariacoalfield
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Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109 101
3 Multivariateregressionanalysis cancapableforsolvingcomplexproblemshasresultin
Artificial neural network. It function as similar to a
Multivariateregressionanalysisisastatisticalmethod human brain works and architecture is complex like
used in estimation the relationship between a depen- anyhumanneuralnetwork.Neuralnetworkareusedto
dent variable and one or more independent variables learnfromthelargedataset.
(Alexopoulous2010).Itcomprisesofmanytechnique ANN consists an input layer, output layer/Target
for analysis and modelling of the dependent and layerandoneormorehiddenlayers(Cilimkovic2015;
independent variables. In other words, the regression Santos et al. 2013). Input Layer, hidden layers and
analysispurposeittocalculateYonthebasisofXor outputlayersorTargetlayerareconnectedtoeachother
todefinethedependencyofYonXi.e. bynodes.Numberofhiddenlayersandnodesvaryfrom
therequiretrainingandresultrequired.Everyconnec-
X ; X ;...;X )Y
1 2 k
tion has some weight associated with them. Input
The X (X , X , …, X ) are ‘‘independent’’ variable, parameters whose value remain same throughout the
i 1 2 k
whileYisa‘‘dependent’’variable. networkareassignedtoinputlayer.Hiddenlayeraccept
A Linear regression is made to estimate the the data from input layer which are send down to the
coefficientsofthelinearequationwhichareinvolving nodes in hidden layer. Hidden layer uses input values
one or more independent variables so that best from input layer and modify them using some weight
prediction of the value of the dependent variable can valueandthennewvaluesaretransferredtotheoutput
bemadequantitatively. layer where it also modify by some weight from
In the multivariate linear regression model, Y has connection between hidden and target layer. Target
normaldistributionwithmean layer process the values from the hidden layer and
produce new values called as output. After that, these
Y¼b þb X þ(cid:2)(cid:2)(cid:2)þb X þrð(cid:2)Þ
0 1 1 q q outputareprocessedbyactivationfunctiontogivethe
The parameters b þb þ(cid:2)(cid:2)(cid:2)þb and r are esti- results.Thenonlinearityisintroducebytheactivation
0 1 q
matedfromdata. function.Therearethreetypesofactivationfunction(1)
linear function (2) threshold function and (3) sigmoid
b ¼intercept
0
function.Inthispaperwehaveusesigmoidactivation
b ...b ¼regressioncoefficients function. The architecture can be seen in following
1 q
figure.
r¼r ¼residualstandarddeviation Output ‘y’ can be calculated, if each neuron has x
res
inputsbyequation
IntheequationY¼b þb X þ(cid:2)(cid:2)(cid:2)þb X ;b is
0 1 1 q q 1
the mean increase in Y per unit increase in X, when !
i n
X
otherX’sarekeptfixed.Ingeneral,b isinfluenceof y¼f ðw (cid:3)x þbÞ
i i i i
Xi corrected (adjusted) for the other X’s. This i¼1
estimationmethodfollowstheleastsquarescriterion.
wherex aretheithinput,w aretheithweight,bisthe
i i
If b , b , …, b are the estimates of b , b , …, b
0 1 q 0 1 q biasandfistheactivationfunctionfortheneuron.
thenthe‘‘fitted’’valueofYis
The new information to be obtain was get only
Yfit ¼b0þb1X1þ(cid:2)(cid:2)(cid:2)þbqXq when the network is being trained. In this paper we
haveusebackpropagationalgorithmwhichisthemost
The b , b , …, b are computed such that
0 1 q robust and versatile technique in all algorithm. It
R(Y - Y )2tobeminimal.Since(Y - Y )iscalled
fit fit results in efficient learning method for multilayer
theresidual;onecanalsosaythatthesumofsquared
perception(MLP)neuralnetworks.
residualsisminimized.
Whenthereceivedinputsareforwardedthroughthe
all others next layer to obtain the output then it is
4 Backpropagationneuralnetwork known as feed forward back-propagation neural
network (BPNN). The learning capability of back
To develop the understanding on functioning of propagation networks depend upon the internal map-
human brain and desire of building a machine that pingofthecharacteristicsignalfeaturesintheprocess
123
102 Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109
ofnetworktraining ontothehiddenlayer.Thestored Table2 Ranges of the input and output parameters used to
mappingsinthislayerduringthetrainingphaseofthe developintelligentmodels
networkgetsautomaticallyretrievedduringitsappli-
Parameters Unit Range SD
cationphase.
Input
When the network is trained, data processed from
input to hidden layer and hidden to output layer. At Moisture % 0.5–70 2.26
output layer, the output is compared to the measured Volatilematter % 32–45 3.44
values (targetvalues)andthe error isprocessedback Ash % 10–48 9.92
throughthenetworkbyupdatingweightsandbiasesof Fixedcarbon % 18.2–45 6.92
distinct neurons. The network is trained until mini- Vitrinite % 0.2–86.1 32.90
mum network error is reached (Verma and Sirvaiya Semi-vitinite % 0–2.30 0.73
2015).Theerror is calculated fromroot mean square Litinite % 0–12.8 4.56
error(RMSE). Intertinite % 5.5–64.6 22.31
In the paper, transfer function are nonlinear Mineralmatter % 4.5–81.2 20.37
sigmoid function (LOGSIG, TANSIG) and linear Mean Ro% 0.61–1.94 0.35
function (POSLIN, PURELIN). The logarithmic sig- Depth M 46–996 309.25
moidfunction(LOGSIG)isdefinedas Target
Langmuirvolumeconstant mL/gm 18–89 20.5
1
f ¼ Langmuirpressureconstant kPa 0.8–2.732 0.66
1þe(cid:4)ex
whereas the tangent sigmoid function (TANSIG) is
definedas[7]
eex (cid:4)e(cid:4)ex
f ¼
eex þe(cid:4)ex
where e is the weighted sum of the inputs for a
x
processingunit.
The sample values of proximate and macerals
parametersaretakenasinputdataandsamplevalues
ofLangmuirvolumeconstantandLangmuirpressure
constant of CO are consider as target data (Table 2;
2
Fig. 3).
5 Generalizedregressionneuralnetwork
Fig.3 Aschematicdiagramofneuralnetwork
AGeneralizedRegressionNeuralNetwork(GRNN)is
a variation of the radial basis neural networks which InPatternlayereachneuronspresentstrainingpattern
does not require an iterative training procedure like andoutputfromthelayer.Summationlayerisnextto
back propagation networks needed (Hannan et al. the pattern layer. Summation layer with the output
2010).Hiddenorkernelregressionnetworksisabasis layer perform a normalization of output set. Radial
of GRNN. Any arbitrary function between input and basisandlinearactivationfunctionsareusedinhidden
output vectors are approximated by GRNN. As the and output layers while training the network. The
trainingsetseizetendstoahighervalue,thefunction structureofGRNNisshowninFig. 4.
estimation error approaches to zero, with minor The predicted value Y’ generated from an
i
constraintsonthefunction. unknowninputvectorxcanbegivenbyequation:
GRNN has input layer, pattern layer, summation
layerandoutputlayerasinfigureThenumberofthe Y0 ¼Pni¼1yi(cid:2)expð(cid:4)Dðx;xiÞÞ;
observation parameters define number of input units. i Pn expð(cid:4)Dðx;xÞÞ
i¼1 i
123
Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109 103
Theoptimizationoftheparametersassociatedwiththe
RBFoccursinnetworktraining.Whentrainingnetwork
are accurately assumed then linear combination of
RBFs can give no error at the training vector. This
fitting of RBFs to data for function approximation is
closelyrelatedbydistanceweightregression.
MoodyandDarken(1989)hasgivenaradialneural
network in which they have selected exponential
activation function for their radial basis function
networks. The exponential activation function is
similartotheGaussiandensityfunctioncenteredatc.
i
!
jjx (cid:4)cjj2
F ¼exp (cid:4) i i
i r2
i
wherethefunctionspread(r)aroundthecentrefinds
i
the ratio of the function decay with its distance from
Fig.4 Aschematicdiagramofgeneralizedneuralnetwork
the centre. Spread constant value should be selected
Dðx;xÞ¼Xm (cid:2)xi(cid:4)xik(cid:3)2 small enough in order to restrict the basis function
i
r spreading(PalitandPopovic2005).
k¼1
where y is the weight connection between the ith
i 7 Layerrecurrentneuralnetwork
neuron in the pattern layer, n is the number of the
trainingpatterns,DistheGaussianfunction,misthe
Alayerrecurrentneuralnetwork(LRNN)isthattypeof
numberofelementsofaninputvector,x andx are
k ik
artificial neural network where connections between
thejthelementofxandx,respectively.
i
unitsformadirectedcycle.Thisresultsinaninternalstate
of the network which permits it in exhibiting dynamic
6 Radialbasisneuralnetwork temporal behavior. In this network bi-directional data
flowoccurs.RNNsusetheirinternalmemorytoprocess
RBF networks are similar to generalized regression arbitrary sequences of inputs which is different from
neural networks. In generalized regression neural net- feedforward neural networks. There are feedback con-
works, there have one neuron for each point in the nectionsbetweenunitsofdifferentlayersinRNN.This
training file, whereas RBF networks have a variable explainthatthedependencyofoutputofthenetworkon
numberofneurons.ThesenumberofneuroninRBFis bothexternalinputsaswellasonthestateofthenetwork
usually much less than the number of training points intheprevioustimestepasshowninFig. 5.Themodel
(Hannanetal.2010). explainfullfeedbackemploymentandinterconnections
RBFNcomprisesthreelayers(1)aninputlayer(2) between all nodes. Advantages like the capability to
hiddenlayerand(3)anoutputlayer.Allthelayersare retainvaluesfrompreviouscyclesofprocessing,which
fully connected by the nodes. The input layer has can be used in current computations allows RNNs to
assigned input parameters, and hidden layer neurons produce complex, time varying outputs in response to
compriseGaussiantransferfunctionswhichisusedas simplestaticinputs(Nkoana2011).
radialbasisfunction.Theoutputsfromhiddenlayerare
inverselyproportionaltothedistancefromthecenterof 8 Resultanddiscussion
the neuron. Input Layer are direct to the hidden layer
withoutanyweights.ThetransferfunctionusedisRBF 8.1 Multivariateregression(MVRA)analysis
which is symmetrical about a given mean or center topredictLVCandLPC
point in a multidimensional space. The number of
hidden nodes with RBF activation function in RBFN The macerals parameters are taken as ‘y’ variable
are connected in a feed forward parallel architecture. (independentvariable)andLVC/LPCaretakenas‘x’
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104 Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109
Fig.5 Asimplerecurrent
network
variable (dependent variable) to develop MVRA Hezarkhani 2012; Ahmadia et al. 2013; Verma and
model. A correlation has been generated for each Singh2013;SinghandVerma2010;VermaandSingh
Langmuirconstantwiththegivenmaceralsparamters 2009;SinghandVerma2011).
(Moisture, Volatile matter, Ash, Fixed carbon, Vitri- In this paper, different type of neural network are
nite, Semi-vitinite, Litinite, Intertinite, Mineral mat- considered to predict LVC and LPC governing the
ter,MeanRo%andDepth). nature of Langmuir isotherms which associate the
MultivariatelinearequationforeachLVCandLPC adsorptionofcarbondioxidegasmoleculesonacoal
areobtained.Theequationsare: surface to gas pressure or concentration at a fixed
temperature. The architect of all different neural
LVC ¼ð4:553826842(cid:3)mÞþð0(cid:3)vÞ
networkareoptimallyoptimized.
þð(cid:4)0:627388513(cid:3)aÞþð1:345670812(cid:3)fÞ
ANN application has some issues also and the
þð2:603845223(cid:3)kÞþð17:11085324(cid:3)sÞ most common one is to define optimum number of
þð2:997470493(cid:3)lÞþð1:931219754(cid:3)iÞ hiddenlayers,thenumberofneuronsinthesehidden
layers,functionalrelationsbetweeninputandoutput
þð3:490123182(cid:3)zÞþð22:0153635(cid:3)wÞ
parameters, learning algorithm and avoiding over-
þð(cid:4)0:030223317(cid:3)dÞ
fitting.So,differentneuralnetworkhasbeenconsid-
ered in study and a simulation has been done on the
And
sameproblem.
LPV ¼ð0:119074233(cid:3)mÞþð0(cid:3)vÞ TheRMSEmethodhasbeenusedtodeterminethe
optimal model parameters. The RMSE is more
þð(cid:3)(cid:4)0:039505654(cid:3)aÞþð0:030893488(cid:3)fÞ
sensitive to the larger relative errors which are
þð0:090233506(cid:3)kÞþð0:404806961(cid:3)sÞ
resulted due to the low valued so that it offers a
þð0:09012357(cid:3)lÞþð0:084394968(cid:3)iÞ balanced evaluation of the goodness of fit of the
þð0:112784163(cid:3)zÞþð(cid:4)0:213303844(cid:3)wÞ model. The perfect model will have a RMSE value
þð(cid:4)0:000263957(cid:3)dÞ approachingtozero.Theformulafordeterminingthe
RMSEis:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8.2 Artificialneuralnetwork(ANN) ðO (cid:4)TÞ2
RMSE ¼ i i
N
Several researchers have used ANN to predict rock
parameterslikeoilflowrateofreservoir,permeability, where, Ti is measured output (Target),Oi is the
dynamicelasticconstants,compressivestrength,creep predicted output and N represent the number of
parameters, ground vibration etc. (Tahmasebi and input–outputdatapairs.
123
Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109 105
Fig.6 OptimumANNNetworkwithtypicalfeed-forwardbackpropagation
Table3 ComparisonofthedifferentANNarchitecturebased ‘logsig–logsig-purelin’ has been found to have the
onRMSEvalues
minimumRMSEofvalue0.0022(Fig. 6;Table 3).
S.no. Transferfunction Model RMSE
8.2.2 Generalisedregressionneuralnetwork(GRNN)
1. Tansig-purelin 11-7-2 3.4058
2. Tansig-purelin 11-20-2 3.0854
The generalized regression neural network is consid-
3. Logsig-purelin 11-20-2 2.7874
ered and input parameters are assigned to input data
4. Tansig–tansig-purelin 11-15-5-2 0.3633
and measured LVC and LPC value are assigned to
5. Logsig-logsig-purelin 11-15-10-2 0.0022
target data. Input has been simulated with different
6. Logsig-poslin 11-15-2 6.2289
spread constant of 1, 5, and 10. The outputs are
7. Tansig-tansig-purelin 11-7-15-2 1.5198
obtained and RMSE value has been calculated from
8. Tansig-tansig-poslin 11-15-20-2 4.1952 obtainedoutputtostudytheoptimumofthenetwork.
9. Logsig-logsig-purelin 11-10-5-2 0.7368 Thenetworkviewforgeneralisedneuralnetworkisas
10. Tansig-logsig-purelin 11-10-20-2 4.8785 follows(Fig. 7;Table 4).
8.2.1 Backpropagationfeedforwardneuralnetwork 8.2.3 Radialbasisneuralnetwork(RBNN)
(BPNN)
The radial basis network is considered and input
To determine the optimal architecture, two and three parameters are assigned to input data and measured
hidden layer in network has been considered for LVCandLPCvalueareassignedtotargetdata.Input
carrying out parametric simulation. In each hidden hasbeensimulatedwithdifferentspreadconstantof1,
layer, number of neurons has been changed to 10, and 100. Output for both Langmuir isotherm
determine the optimum model based on minimum constant has been obtained. There is no change in
valueofRMSE.The80 %datasetisusedfortraining, output with the change in value of spread constant.
10 %forvalidationand10 %fortesting.Thenetwork Thenetworkviewforradialbasisneuralnetworkisas
with architecture 11–15–10–2 with transfer function follows(Fig. 8).
Fig.7 OptimumGRNN
network
123
106 Geomech.Geophys.Geo-energ.Geo-resour.(2016)2:97–109
8.2.4 Layerrecurrentneuralnetwork(LRNN) Table7 RMSEofdifferentpredictionmodelsforLPC
Model Rootmeansquare
Inputdataaresimulatedwithdifferenttrainingfunction
error(RMSE)
and adaption learning function in Layer recurrent
neural network. The training function considered are MVRA 8.1305
TRAINLM, LEARNGDM and TRAINLM whereas BPNN 0.0970
adaptionlearningfunctionconsideredareLEARNGDM GRNN 0.00064
and LEARNGD. The performance function is MSE. RBNN 2.3e-23
ThenumberofLayersare2with15neuronin1stLayer LRNN 0.00443
and transfer function as logsig-purelin. The network
viewforrecurrentneuralnetworkisas(Fig. 9;Table 5):
The optimized model is for network with training
function = TRAINLM, Adaption Learning Funtion Table6 RMSEofdifferentpredictionmodelsforLVC
Model Rootmeansquare
error(RMSE)
Table4 Comparison of the different GRNN architecture
basedonRMSEvalues MVRA 279.6838
BPNN 1.2570
Spreadconstant RMSE
GRNN 0.0140
5 0.0099 RBNN 1.4e-20
10 0.4431 LRNN 1.23
Fig.8 OptimumRB
network
Fig.9 Optimumrecurrent
network
Table5 RMSEvaluesfor
S.No Trainingfunction Adaptionlearningfunction Performancefunction RMSE
differentnetwork
1 TRAINLM LEARNGDM MSE 3.346640106
2 LEARNGDM LEARNGDM MSE 16.73320053
3 TRAINLM LEARNGD MSE 4.949747468
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