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SPE 562  
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ANALYSIS OF GAS WELL BEHAVIOR USING’  
A TWO-DIMENSIONAL  
UNSTEADY-STATE  
MODEL  
By  
O. G. Kiel, Jr. Member AIME, Continental0?.1Co., Ponca cityjOkla. and  
John M. Campbell,MemberAIME, Univ. d? Okl.a.,Norman,Okla.  
PublicationRightsReserved  
This paperis to be presentedat the OklahomaUniversity-SPEProductIon-ResearcShymposiumin  
Norman,Okla.,on April 29-30, 1.963,and is consideredthe propertyof the Societyof PetroleumEngi-  
neers. Permissionto publishis herebyrestrictedto an abstractoif not more than 300 words,with no  
illustrations,unlessthe paper is specificallyreleasedto the pressby the”Edi’orof the Journalof  
.
PetroleumTechnologyor the ExecutiveSecretary.  
ment of where and by whom the paper is presented.  
Such abstractshculdcon%ainconspicuousacknowledge-  
Publicationelsewhereafter publicationin Journal  
of PetroleumTechnologyor ~ocietyof PetroleumEngineersJournalis grantedon request,providing  
propercredit is given that publicationand the originalpresentationof the paper.  
Discussionof this ~amr is invited. Three copiesof anY discussionshouldbe sent to the Society  
.-  
of PetroleumEngineersoffice. Such discussionmay ;e present&1at the above meetingand considered -  
for publicationin one of the two SPE magazineswith the paper. ‘  
accuratelyanalyzeresultsobtainedtiuringthis  
periodusing one-dimensionaltechniques. The  
lerigth”otfhis intervalcan be a matter of min.  
‘utesor hours for ttin permeableformations,or  
days in”khecase of somethick low-permeability  
reservoirs. Specifically,tlietime duration  
dependsupon the completionLnterval,formation  
thickness,wellstimulation,horizontal’and  
verticalpermeabilitydistributions,and porosity.  
KBSTRACT  
A finiteclifferenca~pproximationto the two-  
dimensionalpartialdifferentialequatiocfor un-  
steady-stategas flow that Ls capableof describing  
flow behaviorthroughwidelyvaryingpermeability  
Additionally,  
configurationshas been developed.  
it is capableof describingflow throughvarious  
types of boundaryconditionssuch as open-hole.  
completions,partiallypenetratingwells and  
single-planefractures. Inclusionof the vari-  
able flow rate boundaryequationsmakes it possi-  
ble to studyperiodsof pressurebu%ld.upas well  
as draw-downtests. Whilethe treatmentof the  
The successionof steady-stateapproximation  
method has been simwnto give reliablepredictions  
when using properlyselecteddata.  
This means  
that routinecomputationalmethodscan be used  
for reliablepredictivecalculationsif the data  
from a meaningfult,est are properlyused. Gen-  
eral criteriafor this selectionof test and  
permeabilityfunctionsis not entirelycorrect  
as verlf$edby the inc]’easinmgaterialbalance  
error for permeabilitydiscontinuitiesat large  
distancesaway from the wellbore,it does provide  
a much greaterdetaileddescriptionof *he possible  
permeabilityvariationsthan heretoforepresented.  
data have been included.  
Basically,these depen~  
on the durationof the verbicalcomponentof flow  
to be expected.  
Throughuse of the model we have established INTRODUCTION  
-
the existenceof a relativelyshorttransientperi-  
od followingeach changein flow rates f,orall  
geometriesotherthan uniformpermeabilitywith  
There has been considerableeffortin recent  
years on the developmentof mathematicalreservo~r  
an open-holecompletion, This periodconstitutes models. This has involvedthe use of one.dimen-  
sionalmodelswhich couldbe used to.!tfitdata ‘  
thetime requiredfor the pressuregradientstu”  
traversethe distancefrom the perforatedinter- ,  
val or fractureplane to both the upper and lower  
boundariesand to establtshflow paths such that  
the entireverttcaldistributionof reserves  
contributesits proportionates@re of t.he’pro-  
Iuction. TheoreticallyjTt is not possibleto  
.,  
* Based on work submittedfor DoctoralDisser-  
$ation,Universityof Oklahoma,Schoolor  
PetroleumEngineering.  
, ‘“  
.,  
.