1. Artificial Neural Network to Identify Vertical
Fractured Wells Flow Period
Directed Research
Long Vo
9/13/2015

2. 1 UNIVERSITY OFSOUTHERN CALIFORNIA
Contents
1-1 Abstract..................................................................................................................................2
2-1 Introduction............................................................................................................................2
3-1 Back Propagation Neural Network...........................................................................................1
Neural Network Level Classes..........................................................................................................3
Node Number Selection..................................................................................................................3
Data Normalization.........................................................................................................................3
Neural Network Training.................................................................................................................4
4-1 Experimentation and Verification............................................................................................5
5-1 Conclusions.............................................................................................................................6
6-1 Nomenclature.........................................................................................................................6
7-1 Subscripts...............................................................................................................................6
8-1 References..............................................................................................................................6

3. 2 UNIVERSITY OFSOUTHERN CALIFORNIA
Artificial NeuralNetwork to Identify VerticalFractured Wells Flow
Period
Vo, Long
1-1 Abstract
A modifiedpatternrecognitiontechniqueusing
the artificial neural networkisproposedto
reduce the non-uniquenessof hydraulic
fracturingmodel selection.A neural network
level systemof differentcomprehensive neural
networksgroupedintothree stagescanhelp
filterandidentifymodelsthata single
comprehensive neuralnetworkcouldnot.This
can help reduce the time and increase the
accuracy of model selection.
2-1 Introduction
Pressure-transienttestingreliesonasignal of
pressure vs. time;thissignal isexaminedby
plottingwithaspecializedfunctionand
analyzingtodetermine the well-test
interpretationmodel.Traditionallywell-test
interpretationmodelswere identifiedbyusing
inverse theoryregression analysisasnotedby
Al-Kaabi andLee1
;howeverthese theorieslack
the capacity to identifythe correctmodel due
to similarsignals producedfrommultiple
models.Toidentifythe correctmodel,an
interpreterisneededwhenanalyzingwelltest
data to qualitativelyselectthe mostrelated
reservoirmodel. Manypossible analytical
modelscanbe investigatedtofindthe best
interpretation model asnotedbyJuniardi and
Ershaghi2
.
Confusion is caused by non-unique signals when
selecting the most related reservoir model. For
hydraulically fractured wells, the need to
identify the flow regime is necessary. To select
the proper flow regime for hydraulically
fractured wells, Cinco-Ley and Samaniego-V3
proposed the use of type curves of
dimensionless pressure and dimensionless time
to analyze the four flow periods on log-log plots
of a vertically fractured well; fracture linear
flow, bilinear flow, formation linear flow, and
pseudo-radial flow. Yet this approach is limited
by the number of presented flow regimes and a
number of controlling parameters. Future flow
model regime identification can cause an
increase in non-uniqueness such as transition
flow period between bilinear and linear flows
and bilinearflowwithwellbore storage effect.
Application of an expert system in model
interpretation through pattern recognition
using the artificial neural network has been
conducted by Ershaghi et al.4
and Al-Kaabi and
Lee1
. This system has made model identification
faster and more accurate as complexity
increases. The expert system imitates the
reasoning process of a human interpreter as
noted by Juniardi and Ershaghi2
. It uses pattern
recognition of back propagation neural network
to identify the well-test interpretation model.
The back propagation neural network is noise
insensitive, and can analyze complex models
witha multitude of responses.
The purpose of this paper is to describe the
research results on how to train a neural net
simulator to identify hydraulically fractured
flow regimes discussed in Cinco-Ley et al3
. and
to extend the previous studies from Al-Kaabi
and Lee1
, and incorporate Ershaghi et al.4
’s

4. 1 UNIVERSITY OFSOUTHERN CALIFORNIA
training and data normalizing methods while
utilizing single back propagated neural
networks in Juniardi and Ershaghi2
. This paper
will also discuss the strengths and weaknesses
of the neural network models level system in
model interpretation and provide model
selectionverification.
3-1 Back PropagationNeural
Network
The use of Neural Networksinwelltest
interpretationwaspresentedbyAl-Kaabiand
Lee1
in1993 to train a neural netsimulatorto
identifythe well-testinterpretationmodel from
the derivative plot.Thismethodeliminatesthe
needforpreprocessingandwritingcomplex
rules.Itis automaticinmodel selectiononly
and cannotestimate model parameters.
Juniardi andErshaghi2
extendedthisstudyto
incorporate the strengthandweaknessesof the
neural networkmodel interpretationthrougha
single backpropagationneural network while
Ershaghi et al.4
extendedthe researchtowards
a hybridapproachof multiple neuralnetworks.
Thismethod,however, cannotdistinguish
betweensimilarpatternsaswell asa
comprehensive neuralnetwork.The hybrid
methodcannotrecognize patterns thatwere
not trained andmisidentifypatternsif not
properlyfiltered.
In thisstudy,backpropagationneural network
modelswere developedtoidentifywell-test
interpretationmodelsfrom hydraulically
fracturedwells.The modelswere identified
usinglog-logtype curve ondimensionless
pressure andtime as notedbyCinco-Leyand
Samaniego-V3
.A levelclasssystemof neural
networkswascreatedtoensure correct
classificationof all modelsconsidered.
It will focusonflowregime recognitionof
hydraulicallyfracturedwells withafinite-
conductivityvertical fracture.The neural
networkwill be usedasa patternrecognition
tool to determine the bestpossible flowregime
fromthe inputdata.Because the flow regime
variesbasedonitsparameters,the shape will
alsovary. Similarpatternswillalso emerge.
Therefore,asmore flow regimesare
discovered,the needforafast expertsystemis
requiredfora fastercomputation,reducingthe
time of analysis.The five flow regime model
usedinthisstudywill be;bilinearflow,bilinear
flow withwellbore storage effect, earlylinear
flow,formationlinearflow,and pseudo-radial
flow. Each flow period ismodeledbasedonthe
referencedformulasfromCinco-Leyand
Samaniego-V3
.The flow periodconfiguration
can be seenin Figure 1 andthe formulasof
each model canbe seeninTable 1.
The networksystemconsistsof three level
classes.The firstclassconsistsof a single
comprehensive neuralnetwork trainedto
recognize anddistinguishall flow regime
modelsgiven.Byusingasingle comprehensive
neural network,similarities of responsesamong
differentmodelscancause the networkto
outputactivationnumberof > 0.2 for similar
models. Therefore,asecond level of neural
networks iscreatedtorecognize the differences
inthese models,all modelswithsimilar
patternswithactivation number>0.2 are
groupedandtrainedinindividualneural
networksinthe secondlevel.If the secondlevel
of neural networkswere notable todistinguish
the patterns,a thirdlevel wouldbe created (see
Figure 2).A detail descriptionof the neural
networkcan be foundin Ershaghi etal.4
,
Juniardi etal.2
,andAl-Kaabi etal.1
The processof trainingthe Back Propagation
Neural Networkforcomplex pattern
recognitionconsistsof generatingweight
factors to fitthe correct model recognition.The
weightsare generatedthroughoutthree layers
of nodeswithin the neural networkasshownin

5. 2 UNIVERSITY OFSOUTHERN CALIFORNIA
Figure 3; the inputlayeri,the hiddenlayerj,
and the outputlayerk. The nodesinthe layers
are connectedby links;the linksprovide apath
of propagationfromlayeri to j and from j to k
withweightfactorsgeneratedineachlinkas
mentionbyAl-Kaabi andLee1
.The networkis
trainedinan iterative method throughback
propagationbyminimizingthe errorbetween
each weightchange until the weightscan
correctlyidentifythe modelsatacertain
activationnumber.Anexampleof the process
can be seeninJuniardi andErshaghi2
’spaper.
The activationnumberiscontrolledinthe
hiddenlayerandoutputlayerbya squashing
sigmoidfunction.If the node isinactivethe
activationnumberwouldbe 0and if the node is
veryactive the activation wouldbe 1,as
mentionbyErshaghi etal4
.The calculationof
the activationfunctioninasingle node inlayerj
can be seenbelow:
𝑂𝑢𝑡𝑝𝑢𝑡𝑗 =
1
1 + 𝑒𝑥𝑝(−𝑇𝑜𝑡𝑎𝑙𝑗)
𝑂𝑢𝑡𝑝𝑢𝑡𝑗 = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑓𝑟𝑜𝑚 𝑙𝑎𝑦𝑒𝑟 𝑗
𝑇𝑜𝑡𝑎𝑙𝑗 = 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑝𝑢𝑡 𝑠𝑖𝑔𝑛𝑎𝑙
𝑡𝑜 𝑛𝑜𝑑𝑒 𝑗 𝑓𝑟𝑜𝑚 𝑎𝑙𝑙 𝑛𝑜𝑑𝑒𝑠
𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑝𝑢𝑡 𝑙𝑎𝑦𝑒𝑟 𝑖
𝑇𝑜𝑡𝑎𝑙𝑗 = ∑ 𝑤𝑖𝑗 𝑂𝑢𝑡𝑝𝑢𝑡𝑖
𝑖
𝑤𝑖𝑗 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 𝑡𝑎𝑘𝑒𝑛 𝑓𝑟𝑜𝑚
𝑎 𝑙𝑖𝑛𝑘 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑙𝑎𝑦𝑒𝑟 𝑖 𝑎𝑛𝑑 𝑗
𝑂𝑢𝑡𝑝𝑢𝑡𝑖 = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑓𝑟𝑜𝑚 𝑙𝑎𝑦𝑒𝑟 𝑖 𝑜𝑟 𝑖𝑛𝑝𝑢𝑡
𝑓𝑟𝑜𝑚 𝑙𝑎𝑦𝑒𝑟 𝑖
Since layeri is the (first) inputlayer,the
weights, andoutputsfromthe sigmoidfunction
are the same as what wasinputintolayeri.The
same processisusedfor layerk withinput
signalscomingfromj nodes.
Before the dataare usedas inputsforlayeri,it
isfirstnormalizedforthe x-axisandy-axis.
Because the outputfromthe sigmoidfunctionis
from0 to 1 the (x,y) pairsfrom the type curve
plotswere normalizedfrom0.1 to 0.9. At
positive infinity,the sigmoidfunctionwill be 1
and at negative infinity,the sigmoidfunction
will be 0.
0.1 ≤ ( 𝑡𝐷) 𝑜𝑟 ( 𝑡) ≤ 0.9
0.1 ≤ ( 𝑝𝐷) 𝑜𝑟 ( 𝑝) ≤ 0.9
𝑡𝐷 = 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠 𝑡𝑖𝑚𝑒
𝑡 = 𝑡𝑖𝑚𝑒
𝑝𝐷 = 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑝 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
The normalizationwasdone bythe scaling
downmethod:
𝑥 𝑛𝑜𝑟 = 0.1 + 0.8 ×
(𝑥−𝑥 𝑚𝑖𝑛)
𝑁𝑥
𝑦𝑛𝑜𝑟 = 0.1 + 0.8 ×
(𝑦−𝑦 𝑚𝑖𝑛)
𝑁 𝑦
𝑥 = log( 𝑡𝐷) 𝑜𝑟 log(𝑡)
𝑦 = log( 𝑝𝐷) 𝑜𝑟 log(𝑝)
𝑥 𝑚𝑖 𝑛 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑙𝑜𝑔𝑜𝑟𝑖𝑡ℎ𝑚 𝑡𝐷 𝑜𝑟 𝑡
𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡
𝑦 𝑚𝑖𝑛 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑙𝑜𝑔𝑜𝑟𝑖𝑡ℎ𝑚 𝑝𝐷 𝑜𝑟 𝑝
𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡
𝑁 𝑥 = 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓
log 𝑐𝑦𝑐𝑙𝑒𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑡𝐷 𝑎𝑛𝑑 𝑡 𝑎𝑥𝑖𝑠
𝑁 𝑦 = 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓
log 𝑐𝑦𝑐𝑙𝑒𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑝𝐷 𝑎𝑛𝑑 𝑝 𝑎𝑥𝑖𝑠

6. 3 UNIVERSITY OFSOUTHERN CALIFORNIA
For thisstudy,the sectionsbelowanalyze the
processof buildingthe neural networkforwell
testanalysisfollowingthe above-mentioned
methods.
Neural Network Level Classes
The networksystemwasdividedintothree
level classes.Eachclasswas usedto distinguish
patternsoutputfromthe trainedmodelsfrom
the firstlevel throughthe thirdlevel.The first
level classconsistsof earlylinearflow,bilinear
flow,bilinearflow withwellbore storage,
formationlinearflow,andpseudo-radialflow.
Because there are similaritieswithineach
model,the neural networkwouldgenerate
activationnumbershigherthan0.2 forall
patternssimilartothe inputpatterns.Thisis
causedby the non-uniquenessof the patterns.
Therefore,the secondlevelclass of the neural
network systemis createdtodistinguish
betweenthe non-uniquenessoutputtedfrom
level one.
Level twoneural networksconsistof four
groups:
1. Bilinearflow,formationlinearflow,and
pseudo-radialflow
2. Early linearflow,bilinearflow with
wellbore storage, andformationlinear
flow
3. Bilinearflow,bilinearflow with
wellbore storage, andpseudo-radial
flow
4. Bilinearflow,bilinearflow with
wellbore storage, andformationlinear
flow
Howevernotall of the non-uniquenessis
eliminated, therefore, athirdlevelclassneural
networkis createdto eliminate any
misidentification.
Level three neural networksconsistof six
groups:
1. Bilinearflow andbilinearflowwith
wellbore storage
2. Early linearflow andbilinearflow with
wellbore storage
3. Bilinearflow, andformationlinearflow
4. Early linearflow,andbilinearflow
5. Bilinearflow withwellborestorage, and
formationlinearflow
6. Bilinearflow, andpseudo-radial flow
To identify aflow regime, the inputdataenters
the firstlevel neural network;the firstlevel
neural networkoutputseitheraone to three
possible matches.Todistinguishthe non-
uniqueness forthe twoor three outputs,level
classtwo withneural network groupnumber
similartothe outputsfromlevel class,one is
used.The outputfromlevel classtwo can be a
single outputortwooutputs.To distinguish
between the lasttwonon-uniquenessoutputs;
level classthree withneural network group
numbersimilartothe outputsfromlevel class
twois used.
Node NumberSelection
The numberof nodesforthe i inputlayerof
each neural networkconsistsof 100 nodesthat
accept 50 data pointsof (x,y). The hiddenj layer
consistsof 50 nodes;50 nodesisconsistentfor
convergence basedonexperimentation. The
numberof nodesin the outputk layersdepends
on one of the three level classes.The firstlevel
consistsof 5 patternstobe recognized
therefore consistsof 5nodes,the secondlevel
with3 nodes,andthe thirdlevel with2nodes.
Data Normalization
Thisstudyusesthe theoreticallygenerated log-
logtype curve plots’dimensionlesspressure
and dimensionless timedataof hydraulically
fracturedflow regimesasinputsforlayeri. The
minimumlogarithmyandx are determinedby
the characteristicsof each patternandtheir
respective x-axisrange duringtype curve

7. 4 UNIVERSITY OFSOUTHERN CALIFORNIA
matching. To correctlyidentifythe model
duringtype curve matchingthe x-axisrange for
each model needstobe the same,asper Cinco-
Leyet al3
.
Each neural network withinthe neural network
level systemconsistsof anx-axisrange
incorporatingall patterncharacteristicsof all
models.The modelsare fittedwithaminimum
x-axiswithall patternsstartingfromthe same
xmin value, asshownin Figure 4-8 of level one
with50 randomlygeneratedpatterns.Unless
the pattern characteristicof the model extends
past the xmin value anexceptioncanbe made as
longas the numberof cyclesremainsequal to
the rest of the models. The y-axisisdependent
on the x-axisvalues, therefore,cannotbe
selectedindividuallyforeachmodel.The range
will be determinedbythe model withthe
highesty-axislogarithmicvalue atthe highest x-
axisdata pointandthe lowesty-axislogarithmic
value at the lowest x-axisdatapoint.If a
normalizedy value islessthan0.1 causedby
the x-axisdatapoints,0.99 will be used,andif
the ynor is greaterthan0.9, 0.01 will be used.
Level 2 and level 3’sx-axisare chosen similarly
to level one andis basedonthe distinguishable
characteristicsof the patternsto be recognized.
A table of the x-axisandy-axislogarithmic
range of eachlevel canbe seenin Table 2.
Nx and Nyare chosenbasedonthe x-axisandy-
axisrange and are determinedwhenthe full
range are determined toincorporate all pattern
characteristicsof the models.A table of Nx and
Nyfor each level canbe seenin Table 3.
Selectingthe correctx andy range to
incorporate distinguishabledifferences
betweeneach patterncanreduce the number
of type curvesneededtotrainthe neural
networkforall givenreservoirflowmodels.
Doingso will allow the neural networkto
recognize modelsthrough partial datainputs
and shiftingof the partial data,similarlytotype
curve matching.
Neural Network Training
Before the neural networkcanfreelyidentify
reservoirmodels,trainingisrequired.During
the trainingstage,the neural networklearnsto
recognize andseparate differentpatternsby
adjustingthe weightsbetweeneachlayers
usingthe back propagationtechnique
mentionedabove. The trainingsetusedtotrain
the neural networkconsistsof generated
patternsof each model givenbyvarying
parameterswithinanexpectedrange as
mentionedbyErshaghi etal.4
,the minimum
and maximumvaluesof the variedparameters
for eachreservoirflow regimemodelscanbe
seeninTable 4. The parametersare varied
randomlyusingthe Monte Carlosimulationby
rectangulardistribution.
The trainedneural networkcanrecognize an
unknownpatternsimilartothe trainedpattern
witha recognitionlevel usingthe activation
number.Asmentionedthe range usedinthis
paperis from0.1 to 0.9. The higherthe
activationnumberthe higherthe similarity.As
mentionedbyErshaghi etal.4
activationnumber
of greaterthan0.4 is adequate forpattern
recognition.Howeverinthis paperanactivation
of higherthan 0.2 isused fora more
comprehensive networkthatrequiresahigher
numberof trainingsamples.
Trainingthe neural networkcan take a long
periodof time of a few hoursto a few weeks.
However,thiscanbe a limitof the computer
hardware usedto trainthe network;a topof
the line computersystemcancompute the
trainingina few hourswhile a low-end
computersystemcantake up to a few weeks.
Therefore,the trainingof the neural network
shouldbe done ona systemof computers
dedicatedtoneural networkcomputations.
Nevertheless,afterthe networkhasbeen

8. 5 UNIVERSITY OFSOUTHERN CALIFORNIA
trained,the trainedneural networkwill only
take secondsto outputan answer.
The iterative processfortrainingthe neural
networkscanbe seenbelow:
1. 1000 patternsare generatedforeach
flowregime model usingthe Monte
CarloSimulation.
2. The neural networksare trainedforthe
selectedpatterns andflowregime
models.(The numberof flowregimes
groups traineddependsonthe neural
networklevel class.)
3. The trainednetworksare testedwith
10,000 patternsgeneratedfromthe
Monte CarloSimulation foreachflow
regime.
4. Patternswithactivationof lessthan0.2
for each desiredoutput flowregimes
nodesbeingidentifiedare addedtothe
trainingset. Patternswithactivationof
more than 0.2 forflowregimes nodes
not beingidentified are addedtothe
trainingset. Anexample canbe seenin
Figure 9.
5. Steps2 to 4 are repeateduntil there are
no more patternsto addin step4 or
deemedsuitable bythe stopping
algorithm.
Because the networksare trained
comprehensivelyforeachlevelclass,the
numberof outputsdependsonthe numberof
flowregime modelstrainedforthat class,
outputswith activationnumberhigherthan0.2
will be consideredasa possible selection.
Howeverduringthe training process,activation
numberfora specificflow regime higherthan
0.2 not correctlyidentified are alsoaddedto
the trainingset. Addingthese samplestrainthe
neural networktoidentifypatternsthatare
selectedtobe correctand those that are
selectedtobe incorrect.
4-1 Experimentationand
Verification
In orderto verifythe patternrecognition
strengthof the Neural Networks,5sample
patternswere created usingthe Monte Carlo
simulation foreachflow regime model;bilinear
flow,earlylinearflow,formationlinearflow,
pseudo-radialflow,andbilinearflow with
wellbore storage.The x-axisrange selectedwas
between10^-1to 10^5 basedon the level class
one neural network,however, the x-axisrange
can vary basedon the fieldrecordeddata. Level
classone was selectedbecause it isthe first
level of model identification. The y-axisrange
will varybasedoneach flow model.
The data was thennormalizedbasedonthe
range of eachlevel classesmentionedabove
usingthe scalingdownmethod. Theywere used
as inputsforall level classes. Allmodelswere
identifiedcorrectlywithactivationlevel higher
than 0.8 throughthe lastlevel class(3) withall
else lowerthan0.2 as can be seenin Table 5-7.
Thisverifiesthatthe neural networkclass
systemswere able toensure thatthe models
were recognizedthroughpartial datainputsand
shiftingof the partial data. However,itwasnot
able to distinguishbetweenbilinearand
formationlineardistinctively,atlevel 3group 3,
the normalizedrange of the x-axis10^-2 to
10^5 was notable to absolutely distinguish
bilinearfromformationlinearflowwhen
formationlineardatasetwastested. Thiscan
be seenin Table 7, non-uniquenessof the data
was notcompletelyeliminated. When
formationlineardatawere usedasinputsinto
the neural network,the network outputan
activationnumberof lessthan 0.8. Thisproblem
may be due to the fact that formationlinear
flow doesnothave multiple uniquetraining
patterns butconsistsof onlyone.Thiscan cause
the networktomisidentifythe models.This

9. 6 UNIVERSITY OFSOUTHERN CALIFORNIA
problemissimilarlystatedby Juniardi and
Ershaghi2
.
The range selectionduringthe trainingprocess
can alsoplay a role inmisidentification.If the x
and y range of bothbilinearandformation
linearflowwere notselectedtoshowadistinct
difference inpatternbetweenthe two,the non-
uniquenesswillnotbe eliminated. Information
fromothersourcesas mentionedbyErshaghi et
al.4
can be usedtoverifythe chosenmodels.
Data such as one-fourthslopeona log-log
graph mentionedbyCinco-Leyetal.3
toverify
bilinearflow orone-half slopeforlinearflow.
Therefore conventionalmethodswithhuman
verificationsare necessarytoselectthe final
flowmodel whenuncertaintyispresent.
The filteringprocessof the levelsystemswas
thenappliedtotestitsefficiency.The networks
were able toidentifythe desiredflowregime
correctly.The pseudo-radialflowregime, the
bilinearwithwellbore storage flowregime,and
the earlylinearflowregime didnotrequirea
three-stage filtering.A twostage filteringwas
adequate ascan be seenin Figure 10,12, and
14. Bilinearflowandformationlinearflow
requiresathree-stage processtoidentify the
flowregime forthese particularpatterns (see
Figure 11 and13).
5-1 Conclusions
The problemof non-uniqueness hascreateda
needfordecreasingtime andincreasing the
accuracy of flowregime identification.Hydraulic
fracturingmodeling’snon-uniquenesscanoccur
whenmore advancedmodelsof flowregimes
are added,parameterestimationorevenmodel
identificationthroughtype curve matching can
become more complex anddifficulttosolve.
Previouslystudiesof the single comprehensive
neural network helpidentifysimilarpatterns
exhibitedbythe currentmodels.However
cannot distinguishthemwhenthe similarities
are toogreat to separate. Therefore,alevel
classneural networksystemproposedusing
multiple neural networkstrainedtorecognized
and separate the modelsasa formof filtering,
decreasesthe chancesof non-uniqueness.
6-1 Nomenclature
C = wellbore storage coefficient
𝑘 𝑓 𝑏𝑓 = 𝑓𝑟𝑎𝑐𝑡𝑢𝑟𝑒 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦
(𝑘 𝑓 𝑏𝑓) 𝐷 = 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠
𝑓𝑟𝑎𝑐𝑡𝑢𝑟𝑒 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦
𝑝 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑡 = 𝑡𝑖𝑚𝑒
𝑠 = 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑠𝑝𝑎𝑐𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑟′ 𝑤 = 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑤𝑒𝑙𝑙𝑏𝑜𝑟𝑒 𝑟𝑎𝑑𝑖𝑢𝑠
𝑥, 𝑦 = 𝑠𝑝𝑎𝑐𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠
𝑥 𝑓 = 𝑓𝑟𝑎𝑐𝑡𝑢𝑟𝑒 ℎ𝑎𝑙𝑓 − 𝑙𝑒𝑛𝑔𝑡ℎ
η = hydraulic diffusivity
7-1 Subscripts
D = dimensionless
𝑓 = 𝑓𝑟𝑎𝑐𝑡𝑢𝑟𝑒
𝑥 𝑓 = 𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑥 𝑓
𝑤 = 𝑤𝑒𝑙𝑙𝑏𝑜𝑟𝑒
8-1 References
1. l-Kaabi,A.U.,& Lee,W.J. (1993, September
1). UsingArtificial Neural NetworksTo
Identifythe Well TestInterpretationModel
(includesassociatedpapers28151 and

10. 7 UNIVERSITY OFSOUTHERN CALIFORNIA
28165 ).Societyof PetroleumEngineers.
doi:10.2118/20332-PA
2. Juniardi,I. R.,& Ershaghi,I. (1993, January
1). Complexitiesof UsingNeural Networkin
Well TestAnalysisof FaultedReservoirs.
Societyof PetroleumEngineers.
doi:10.2118/26106-MS
3. Cinco-Ley,H.,& Samaniego-V.,F.(1981,
September1).TransientPressureAnalysis
for FracturedWells.Societyof Petroleum
Engineers.doi:10.2118/7490-PA
4. Ershaghi,I.,Li, X.,Hassibi,M.,& Shikari,Y.
(1993, January1). A RobustNeural Network
Model for PatternRecognitionof Pressure
TransientTestData. Societyof Petroleum
Engineers.doi:10.2118/26427-MS

11. 1 UNIVERSITY OFSOUTHERN CALIFORNIA
Table 1
Flow Regime Model
Description
Mathematical Formulations
Type Curve Training and Testing
Data
Model 1:
Early Linear Flow
Model 2:
Formation Linear
Flow
Model 3:
Bilinear Flow
Model 4:
Bilinear Flow with
Wellbore Storage
Model 5:
Pseudo-Radial Flow
𝑤𝐷 Vs. 𝑡 𝐷𝑥𝑓
𝑤𝐷 Vs. 𝑡 𝐷𝑥𝑓
𝑤𝐷 Vs. 𝑡 𝐷𝑥𝑓
( )
( )
𝑤𝐷 Vs.
𝑤𝐷 Vs. 𝑡 𝐷𝑟
Fracture
Fracture
Well
Well Fracture
Fracture
Well
Early LinearFlow
FormationLinearFlow
BilinearFlow
Psuedo-Radial Flow
Fig. 1 Flow Periods for A Vertically Fractured Well (After Cinco-Ley et al.)

12. 2 UNIVERSITY OFSOUTHERN CALIFORNIA
Fig. 3 Set of Weights Obtained After Training The Neural Network (After Juniardi et al.)
BNN Level 1
All Flow
BNN Level 2
BNN Level 3
Group 1 Group 2 Group 3 Group 4
Group 1 Group 2 Group 3 Group 4 Group 5 Group 6
Fig. 2 Neural Network Level Class System Structure

13. 3 UNIVERSITY OFSOUTHERN CALIFORNIA
Fig. 4 Early Linear Flow and Early Linear Flow Normalize
Fig. 5 Bilinear Flow and Bilinear Flow Normalize
Fig. 6 Bilinear with Wellbore Storage Flow and Bilinear with Wellbore Storage Flow Normalize

14. 4 UNIVERSITY OFSOUTHERN CALIFORNIA
Fig. 7 Formation Linear Flow and Formation Linear Flow Normalize
Fig. 8 Pseudo-Radial Flow and Pseudo-Radial Flow Normalize

15. Table 2 Table 3
Table 4
xmin xmax ymin ymax
FlowLevel 1:
Pseudo-Radial -3 3 -2 6
Formation Linear -1 5 -1 7
Early Linear -1 5 -5 3
Bilinearwith Wellbore Storage -1 5 -6 2
Bilinear -1 5 -2 6
FlowLevel 2:
Group 1
Bilinear -1 5 -2 1
Formation Linear -1 5 -1 2
Psuedo-Radial -3 3 -2 1
Group 2
Early Linear -1 5 -5 10
Bilinearwith Wellbore Storage -1 5 -6 9
Formation Linear -1 5 -1 14
Group 3
Bilinear -1 6 -2 6
Bilinearwith Wellbore Storage -1 6 -6 2
Psuedo-Radial -4 3 -2 6
Group 4
Bilinear -1 5 -2 6
Bilinearwith Wellbore Storage -1 5 -6 2
Formation Linear -1 5 -1 7
FlowLevel 3:
Group 1
Bilinear -1 7 -2 6
Bilinearwith Wellbore Storage -1 7 -6 2
Group 2
Early Linear -1 5 -5 4
Bilinearwith Wellbore Storage -1 5 -6 3
Group 3
Bilinear -2 5 -2 3
Formation Linear -2 5 -2 3
Group 4
Early Linear -1 5 -5 4
Bilinear -1 5 -2 7
Group 5
Bilinearwith Wellbore Storage -1 5 -6 2
Formation Linear -1 5 -1 7
Group 6
Bilinear -1 7 -2 3
Pseudo-Radial -5 3 -3 2
Note:Each value is an exponentof base 10.
Flow Level 1:
Nx 6
Ny 8
Flow Level 2:
Group 1
Nx 6
Ny 3
Group 2
Nx 6
Ny 15
Group 3
Nx 7
Ny 8
Group 4
Nx 6
Ny 8
Flow Level 3:
Group 1
Nx 8
Ny 8
Group 2
Nx 6
Ny 9
Group 3
Nx 7
Ny 5
Group 4
Nx 6
Ny 9
Group 5
Nx 6
Ny 8
Group 6
Nx 8
Ny 5
Parameter Minimum Maximum
0 500
0 500
0 10000
(𝑘 𝑓 𝑏𝑓) 𝐷
𝑛 𝑓𝐷
𝐷𝑓

16. Fig. 9 Neural Network Training Set
Filtering
Table 5
FlowLevel1
Early
Linear
Flow
Bilinear
Flow
BilinearFlow
withWellbore
Storage Flow
Formation
Linear
Flow
Pseudo-
Radial Flow
DesiredOutput 0.9 0.1 0.1 0.1 0.1
0.6611 0.051 0.4633 0.0905 0.0935 Yes
0.0152 0.1523 0.5621 0.5854 0.6541 Yes
0.7215 0.0124 0.0152 0.0685 0.0545 No
0.8552 0.0154 0.0752 0.0874 0.0656 No
FlowLevel3:Group4
DesiredOutput 0.9 0.1
0.9016 0.0984 No
0.8154 0.1523 No
0.5452 0.5451 Yes
0.4512 0.8421 Yes
10,000Patterns
OptainedTested
Output
ObtainedTested
Output
AddedTo
TrainingSet
10,000Patterns
Flow Level 1
Early Linear
Flow Bilinear Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.9 0.1 0.1 0.1 0.1
0.6346 0.0573 0.4417 0.1063 0.0934
0.6231 0.0599 0.4355 0.1117 0.0935
0.6335 0.0571 0.4454 0.1048 0.0936
0.6454 0.0531 0.4636 0.0975 0.0934
0.6393 0.0558 0.4488 0.1019 0.0935
Desired Output 0.1 0.9 0.1 0.1 0.1
0.1111 0.3282 0.0946 0.2102 0.5629
0.095 0.3833 0.1534 0.197 0.3916
0.1154 0.3217 0.0966 0.2102 0.5517
0.1049 0.3545 0.1314 0.2023 0.4431
0.1161 0.3244 0.1073 0.2085 0.5121
Desired Output 0.1 0.1 0.9 0.1 0.1
0.5887 0.0273 0.7927 0.0383 0.0957
0.5268 0.0508 0.6112 0.1201 0.0931
0.5235 0.0517 0.6067 0.1229 0.0931
0.5947 0.0219 0.8483 0.023 0.0971
0.5571 0.0414 0.6673 0.0888 0.0936
Desired Output 0.1 0.1 0.1 0.9 0.1
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
Desired Output 0.1 0.1 0.1 0.1 0.9
0.1151 0.3194 0.0998 0.2107 0.5461
0.1158 0.319 0.1015 0.2105 0.5391
0.115 0.3195 0.0997 0.2107 0.5466
0.1153 0.3192 0.1002 0.2107 0.5443
0.1153 0.3192 0.1003 0.2106 0.544
Obtained Tested
Output
Obtained Tested
Output
Obtained Tested
Output
Obtained Tested
Output
Obtained Tested
Output

17. Table 6
Flow Level 2:
Group 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.9 n/a 0.1 0.1
0.6521 0.3336 0.0977
0.9643 0.046 0.0916
0.924 0.0903 0.093
0.9687 0.0408 0.0914
0.9625 0.0482 0.0917
Flow Level 2:
Group 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 n/a 0.9 0.1
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
Flow Level 2:
Group 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 n/a 0.1 0.9
0.11 0.0591 0.9046
0.1066 0.0628 0.9021
0.1102 0.0589 0.9048
0.1094 0.0599 0.904
0.1093 0.06 0.9039
Flow Level 2:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output 0.9 n/a 0.1 0.1 n/a
0.9551 0.042 0.1019
0.9516 0.0456 0.1017
0.8985 0.1007 0.1001
0.881 0.1192 0.0998
0.9022 0.0968 0.1002
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Flow Level 2:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output 0.1 n/a 0.9 0.1 n/a
0.0171 0.9693 0.0989
0.0551 0.9322 0.0925
0.0348 0.9524 0.0931
0.0217 0.9653 0.0958
0.0721 0.9154 0.0925
Flow Level 2:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output 0.1 n/a 0.1 0.9 n/a
0.1018 0.0932 0.9004
0.1018 0.0932 0.9004
0.1018 0.0932 0.9004
0.1018 0.0932 0.9004
0.1018 0.0932 0.9004
Flow Level 2:
Group 3
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.9 0.1 n/a 0.1
0.2762 0.282 0.1413
0.285 0.799 0.1183
0.2829 0.6979 0.1234
0.2849 0.7965 0.1185
0.2846 0.7838 0.1192
Flow Level 2:
Group 3
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 0.9 n/a 0.1
0.285 0.8022 0.1181
0.285 0.8022 0.1181
0.285 0.8022 0.1181
0.285 0.8022 0.1181
0.285 0.8022 0.1181
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Flow Level 2:
Group 3
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 0.1 n/a 0.9
0.2212 0 0.3792
0.2178 0 0.3996
0.2178 0 0.3995
0.2199 0 0.3871
0.2185 0 0.3953
Flow Level 2:
Group 4
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.9 0.1 0.1 n/a
0.6889 0.1083 0.3169
0.6405 0.1272 0.3176
0.68 0.1115 0.3173
0.6504 0.1231 0.3177
0.6655 0.117 0.3177
Flow Level 2:
Group 4
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 0.9 0.1 n/a
0 1 0.0731
0 1 0.0896
0 1 0.0817
0 1 0.0752
0 1 0.0949
Flow Level 2:
Group 4
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 0.1 0.9 n/a
0.6353 0.1292 0.3178
0.6353 0.1292 0.3178
0.6353 0.1292 0.3178
0.6353 0.1292 0.3178
0.6353 0.1292 0.3178
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a

18. 1 UNIVERSITY OFSOUTHERN CALIFORNIA
Table 7
Flow Level 3:
Group 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.9 0.1 n/a n/a
0.8944 0.1056
0.854 0.1456
0.8885 0.1114
0.8634 0.1364
0.877 0.1229
Flow Level 3:
Group 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.1 0.9 n/a n/a
0.0976 0.9024
0.0963 0.9037
0.0965 0.9035
0.097 0.903
0.0963 0.9037
Flow Level 3:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.9 n/a 0.1 n/a n/a
0.9001 0.1
0.8998 0.1003
0.9001 0.1
0.902 0.098
0.9002 0.0999
Flow Level 3:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.1 n/a 0.9 n/a n/a
0.0993 0.9007
0.0996 0.9004
0.0997 0.9004
0.0997 0.9003
0.0992 0.9008
Obtained Tested
Output
n/a n/a n/a
Obtained Tested
Output
n/a n/a n/a
Obtained Tested
Output
n/a n/an/a
Obtained Tested
Output
n/a n/a n/a
Flow Level 3:
Group 3
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.9 n/a 0.1 n/a
0.9008 0.0992
0.8998 0.1002
0.9007 0.0993
0.9 0.1
0.9004 0.0996
Flow Level 3:
Group 3
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.1 n/a 0.9 n/a
0.2953 0.7024
0.2953 0.7024
0.2953 0.7024
0.2953 0.7024
0.2953 0.7024
Flow Level 3:
Group 4
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.9 0.1 n/a n/a n/a
0.901 0.099
0.9007 0.0993
0.9011 0.0989
0.9003 0.0997
0.9012 0.0988
Flow Level 3:
Group 4
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.1 0.9 n/a n/a n/a
0.0996 0.9003
0.1063 0.8939
0.1016 0.8984
0.1055 0.8947
0.1039 0.8962
Obtained Tested
Output
n/an/an/a
Obtained Tested
Output
n/a n/a n/a
Obtained Tested
Output
n/a n/an/a
Obtained Tested
Output
n/a n/a n/a
Flow Level 3:
Group 5
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a n/a 0.9 0.1 n/a
0.8998 0.1002
0.9021 0.0979
0.9016 0.0984
0.9006 0.0994
0.9023 0.0977
Flow Level 3:
Group 5
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a n/a 0.1 0.9 n/a
0.1 0.9
0.1 0.9
0.1 0.9
0.1 0.9
0.1 0.9
Flow Level 3:
Group 6
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.9 n/a n/a 0.1
0.9487 0.0514
0.9458 0.0543
0.9481 0.052
0.9463 0.0538
0.9472 0.0529
Flow Level 3:
Group 6
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.1 n/a n/a 0.9
0.1262 0.8738
0.1396 0.8604
0.1252 0.8748
0.1295 0.8706
0.13 0.87
Obtained Tested
Output
n/an/a n/a
n/a
Obtained Tested
Output
n/a n/a n/a
Obtained Tested
Output
n/a n/a n/a
Obtained Tested
Output
n/a n/a

19. Fig. 10 Early Linear Flow Identification
Fig. 11 Bilinear Flow Model Identification
Fig. 12 Bilinear Flow with Wellbore Storage Flow Model
Identification
Fig. 13 Formation Linear Flow Model Identification
Flow Level 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.9 0.1 0.1 0.1 0.1
0.6346 0.0573 0.4417 0.1063 0.0934
0.6231 0.0599 0.4355 0.1117 0.0935
0.6335 0.0571 0.4454 0.1048 0.0936
0.6454 0.0531 0.4636 0.0975 0.0934
0.6393 0.0558 0.4488 0.1019 0.0935
Flow Level 2:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.9 n/a 0.1 0.1 n/a
0.9551 0.042 0.1019
0.9516 0.0456 0.1017
0.8985 0.1007 0.1001
0.881 0.1192 0.0998
0.9022 0.0968 0.1002
Flow Level 3:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.9 n/a 0.1 n/a n/a
0.9001 0.1
0.8998 0.1003
0.9001 0.1
0.902 0.098
0.9002 0.0999
Obtained Tested
Output
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/an/a
Flow Level 1 Early Linear Flow Bilinear Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.1 0.9 0.1 0.1 0.1
0.1111 0.3282 0.0946 0.2102 0.5629
0.095 0.3833 0.1534 0.197 0.3916
0.1154 0.3217 0.0966 0.2102 0.5517
0.1049 0.3545 0.1314 0.2023 0.4431
0.1161 0.3244 0.1073 0.2085 0.5121
Flow Level 2:
Group 1 Early Linear Flow Bilinear Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.9 n/a 0.1 0.1
0.6521 0.3336 0.0977
0.9643 0.046 0.0916
0.924 0.0903 0.093
0.9687 0.0408 0.0914
0.9625 0.0482 0.0917
Flow Level 3:
Group 1 Early Linear Flow Bilinear Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.9 0.1 n/a n/a
0.8944 0.1056
0.854 0.1456
0.8885 0.1114
0.8634 0.1364
0.877 0.1229
Obtained Tested
Output
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a n/a
Flow Level 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output 0.1 0.1 0.9 0.1 0.1
0.5887 0.0273 0.7927 0.0383 0.0957
0.5268 0.0508 0.6112 0.1201 0.0931
0.5235 0.0517 0.6067 0.1229 0.0931
0.5947 0.0219 0.8483 0.023 0.0971
0.5571 0.0414 0.6673 0.0888 0.0936
Flow Level 2:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output 0.1 n/a 0.9 0.1 n/a
0.0171 0.9693 0.0989
0.0551 0.9322 0.0925
0.0348 0.9524 0.0931
0.0217 0.9653 0.0958
0.0721 0.9154 0.0925
Flow Level 3:
Group 2
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output 0.1 n/a 0.9 n/a n/a
0.0993 0.9007
0.0996 0.9004
0.0997 0.9004
0.0997 0.9003
0.0992 0.9008
Obtained Tested
Output
n/a
Obtained Tested
Output
n/a
Obtained Tested
Output
n/a n/an/a
Flow Level 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow with
Wellbore Storage
Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output 0.1 0.1 0.1 0.9 0.1
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
0.1157 0.3206 0.1069 0.2097 0.5206
Flow Level 2:
Group 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow with
Wellbore Storage
Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 n/a 0.9 0.1
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
0.3343 0.6011 0.1024
Flow Level 3:
Group 3
Early Linear
Flow
Bilinear
Flow
Bilinear Flow with
Wellbore Storage
Flow
Formation
Linear Flow
Pseudo-Radial
Flow
Desired Output n/a 0.1 n/a 0.9 n/a
0.2953 0.7024
0.2953 0.7024
0.2953 0.7024
0.2953 0.7024
0.2953 0.7024
Obtained
Tested Output
n/a
Obtained
Tested Output
n/a
Obtained
Tested Output
n/a n/a n/a

20. Fig. 14 Pseudo-Radial Flow Model Identification
Flow Level 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output 0.1 0.1 0.1 0.1 0.9
0.1151 0.3194 0.0998 0.2107 0.5461
0.1158 0.319 0.1015 0.2105 0.5391
0.115 0.3195 0.0997 0.2107 0.5466
0.1153 0.3192 0.1002 0.2107 0.5443
0.1153 0.3192 0.1003 0.2106 0.544
Flow Level 2:
Group 1
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.1 n/a 0.1 0.9
0.11 0.0591 0.9046
0.1066 0.0628 0.9021
0.1102 0.0589 0.9048
0.1094 0.0599 0.904
0.1093 0.06 0.9039
Flow Level 3:
Group 6
Early Linear
Flow
Bilinear
Flow
Bilinear Flow
with Wellbore
Storage Flow
Formation
Linear Flow
Pseudo-
Radial Flow
Desired Output n/a 0.1 n/a n/a 0.9
0.1262 0.8738
0.1396 0.8604
0.1252 0.8748
0.1295 0.8706
0.13 0.87
Obtained Tested
Output
Obtained Tested
Output
n/a n/a
Obtained Tested
Output
n/a n/a n/a