SPE SPE 22688 Application of the Pulse Decay Technique A. Gilicz, Hungarian Hydrocarbon Inst. SPE Member Copyright 1991, Society of Petroleum Engineers Inc. This paper was prepared lor presentation at the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dalias, TX, October 6-9, 1991. This paper was selected for presentation by an SPE Program Committee foliowing review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is restricted to an abstract 01 not more than 300 words. illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL. ABSTRACT In recent years the pulse decay technique became a frequent tool for the fast and convenient mea- surement of permeability and porosity of mostly tight rocks. The method is superior to conven- tional steady state methods as it is much faster and easier to perform. So far it was applied to linear cores (plugs) only. In the last years radial models appeared widely as they show some ad- vantages over linear models, so the need emerged to apply the method for radial cores. Experimen- tal apparatus and mathematical models have been developed to solve the problem. The current pa- per describes experimental set-up, analytical so- lutions of the governing equation and applications of them to performed measurements. INTRODUCTION As numerous low-quality reservoirs worldwide still present a considerable potential for hydrocarbon production, attention was paid to the investiga- tion of these rock types. Determination of their permeability is usually difficult by the conven- tional steady-state displacement methods because the achievement of the steady-state condition is sometimes questionable, the flow rates are diffi- cult to measure and control, measurements can last for a long time increasing error accumlations. References and illustration at end of paper. 305 To complicate the situation, low permeability rocks are usually sensitive to stress, flow velocities and throughputs, they often contain sensitive clay minerals and fines and are susceptible for demages caused by floods. As under these conditions the measurement of pressure and time is much easier Brace et al. [1] introduced a transient technique, known as the pressure pulse technique for the measurement of permeability. An inlet and an outlet vessel was connected to the core the inlet vessel hav- ing a slightly higher initial pressure as the out- let one. The pressure wave generated in this way was forced through the core and the pressure evo- lution on both ends of the core was followed in time. The log of the pressure difference depicted vs. time gave a straight line the slope of which was proportional to permeability. It was the very first model and the authors made some neglec- tions, which however have influence under other circumstances. This is particularly the relation of the pore volume to vessel volumes [2,3]. Later investigators [4,5] included the core's compressive storage in their solutions. Based on the relation of in- and outlet vessel volumes and the pore vol- ume three experimental setups and solutions exist - they were all treated by the literature:
Transcript
SPE
SPE 22688
Application of the Pulse Decay TechniqueA. Gilicz, Hungarian Hydrocarbon Inst.SPE Member
Copyright 1991, Society of Petroleum Engineers Inc.
This paper was prepared lor presentation at the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dalias, TX, October 6-9, 1991.
This paper was selected for presentation by an SPE Program Committee foliowing review of information contained in an abstract submitted by the author(s). Contents of the paper,as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Societyof Petroleum Engineers. Permission to copy is restricted to an abstract 01 not more than 300 words. illustrations may not be copied. The abstract should contain conspicuous acknowledgmentof where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL.
ABSTRACT
In recent years the pulse decay technique becamea frequent tool for the fast and convenient measurement of permeability and porosity of mostlytight rocks. The method is superior to conventional steady state methods as it is much fasterand easier to perform. So far it was applied tolinear cores (plugs) only. In the last years radialmodels appeared widely as they show some advantages over linear models, so the need emergedto apply the method for radial cores. Experimental apparatus and mathematical models have beendeveloped to solve the problem. The current paper describes experimental set-up, analytical solutions of the governing equation and applicationsof them to performed measurements.
INTRODUCTION
As numerous low-quality reservoirs worldwide stillpresent a considerable potential for hydrocarbonproduction, attention was paid to the investigation of these rock types. Determination of theirpermeability is usually difficult by the conventional steady-state displacement methods becausethe achievement of the steady-state condition issometimes questionable, the flow rates are difficult to measure and control, measurements canlast for a long time increasing error accumlations.
References and illustration at end of paper.
305
To complicate the situation, low permeabilityrocks are usually sensitive to stress, flow velocitiesand throughputs, they often contain sensitive clayminerals and fines and are susceptible for demagescaused by floods.
As under these conditions the measurement ofpressure and time is much easier Brace et al.[1] introduced a transient technique, known asthe pressure pulse technique for the measurementof permeability. An inlet and an outlet vesselwas connected to the core the inlet vessel having a slightly higher initial pressure as the outlet one. The pressure wave generated in this waywas forced through the core and the pressure evolution on both ends of the core was followed intime. The log of the pressure difference depictedvs. time gave a straight line the slope of whichwas proportional to permeability. It was the veryfirst model and the authors made some neglections, which however have influence under othercircumstances. This is particularly the relationof the pore volume to vessel volumes [2,3]. Laterinvestigators [4,5] included the core's compressivestorage in their solutions. Based on the relationof in- and outlet vessel volumes and the pore volume three experimental setups and solutions exist- they were all treated by the literature:
2 APPLICATION OF THE PULSE DECAY TECHNIQUE TO ·RADIAL CORES SPE 22688
THEORY
The variable P is the difference ofP pressure anywhere in the core and the initial core pressure Pini,
i.e.:
The outline of the geometry is depicted in Fig.2.The flow in the core is described with the radialdiffusivityequation:
The applied fluid may be either liquid or gas. Inthe later case thepseudopressure of Al-Hussainyet al.[Il] has to be substituted for pressure andthe adjusted time of Meunier et al.[12] for time.Their deffinitions are respectively:
(2)
(1)
(4)
(3)
02 P 10P p4Jc oP-+--=-or2 r Or k at
P = P - Pini
fP Pm(p) = 2 J.... () (P) dp
P.... P P z
t .. = fic! ~dtpc
also be evaluated. Pressure transducers are located at both upstream and downstream vessels.The model and vessels are located in an air thermostated cabinet to provide thermal stability andreservoir temperatures.
Mter the placement of the core the system is filledup with either gas or liquid, and confining pressure is applied. Reaching equilibrium the valveseparating the core and the upstream vessel - usually that one connected to the centre of the core- is closed and the pressure in this vessel is increased slightly. By opening the valve the pressurewave propagates through the core into the downstream vessel. Pressure evolution in the vessels isrecorded. As the measurement is usually short, itis continued to equilibrium, although this is nota necessary requirement. This is especially usefulin case of large vessels or very low permeabilitieswhere measurement times can get longer.
Pressure data along with geometrical data andviscosity/ compressibility data are the inputs ofcomputer programs which fit the mathematicalmodels elaborated to the measurement data. Twoparameters of these solutions are unknown: permeability and porosity. Programs vary these values in a systhematic way until match of pressuresis achieved.
- inlet vessel volume much less than outletvessel and pore volume [6],
The next stage in the evolution of the methodwas the simultaneous determination of porosityand permeability. The solution was developed byHaskett et al. [8] having numerous advantages: afast method providing two important values fromone measurement uder the same conditions without subjective judgement, etc.
Recently Kamath et al.[9] illustrated that theearly time pressure response of the core in thepulse decay measurement contains even more information, and core heterogenities can be detected.
All former solutions were derived for linear coreplugs. In the last years however radial modelshave been introduced in many labs. Radial modelsuse full diameter cores obtained directly from coring operations in drilling. In radial models (cores)flow of fluids has actually the same radial character as the flow of fluids in the vincity of wells soit is assumed that radial cores represent reservoirflow conditions more accurately.
Also the preparation of radial cores for measurements is somewhat simpler as that of plugs: onehas to cut the top and bottom sides of the coreparalelly and drill a small hole into the centre ofthe core, no special cutting device is necessary,and the core holding is also simpler.
Due to these facts the need emerged to apply thepulse decay technique under radial circumstances.The current paper aimes to present the experimental and theoretical solution of the problem.
EXPERTIMENTALSETUP
- inlet vessel, outlet vessel an pore volume inthe same range [7].
- inlet vessd volume much higher as outletvessel and pore volume [4,5],
The scheme of the experimental setup is shown inFig.!. Prior to experiments the top and bottom ofthe core is cut paralelly and a hole is drilled intothe centre of it. Then it is placed into the cylindrical coreholder. Some silicon paste is placedto the top and bottom side of the core for sealing. The upper side of the coreholder is movable.The effect of overburden pressure can be modelledby providing confining pressure load to this disk.By measuring its shift, rock compressibilities can
306
SPE 22688
Initial conditions are:
A.GILICZ 3
rw < r < R p(r,t = -0) =Pini
Pu(t = -0) = PI
Pd(t = -0) =P2
(5)
(6)
(7)
M 1(r) ={C1P1 [J1(anR) + c:; a,.Jo(anR)] +
C2P2 [ Jl(anr",)-
danJo(anr",)]} Yo(a,.r) (15)
Boundary conditions:
r = rw;t > 0
where
C_ p.cVu
1-2krw 1rH
r = R;t > 0
Pd(t) = p(R; t)
(8)
(9)
(10)
(11)
(12)
M2 (r) = - {C1P1 [C:;anYo(anR)+
Y1(an R) ] + C2P2 [ Y1(an r",)
danYo(anr",)]} Jo(anr) (16)
(17)
P2 is usually equal to Pini , but not necessarily.With this notation however the solution could bekept quite general.
Equations 8 and 11 express that pressure mustbe continous between vessels and core boundaries,whereas equations 9 and 12 express mass conservation i.e. the pressure change rate in the vesselsis proportional to the fluxes into- and out of thecore.
Equation 1 with initial and boundary conditionswas solved by means of the Laplace transform.Details are given in Appendix A. The general solution is:
where
c _ P.CVd2 - 2kR1rH
P(r,t) = Poo+! t M 1(r) + M 2 (r) e-~eC n=1 a~[r",Fl+ RF21
where
(13)
(14)
(18)
and an are the positive roots of the equation
(19)
n = 1,2,3,... ,00
307
4 APPLICATION OF THE PULSE DECAY TECHNIQUE TO RADIAL CORES SPE 22688
Note that P2 is usually zero because P2 = Pini·
This however is not a necessary condition.
Pressures are measured in the vessels, i.e. at r w
and r = R locations, so Eq.14. is needed at thesepoints. If r = rw then after some manipulationsthe numerators become
The general solution includes also particularcases, where the ratio of Vu/Vd is much higheror much less then 1. These cases are not applicable for porosity determination, but some labs mayprefer any of these solutions, so they are providedin Appendix B.
If inlet vessel, pore volume and outlet vessel volumes are in the same range, the solution is sensitive to porosity as well. Figs. 3. and 4. show somesimulated pressure histories. Parameters of theseruns are summarized in Table 1. As can be seenthe rate of pressure changes is a strong function ofpermeability, whereas equilibrium pressures are afunction of porosity.
Early time solutions can be used for heterogenitydetection [9]. In radial system they are as follows:
At r = rw location
APPLICATION
(26)
(25)_-'Lt (fS\P(r1D,t~O)=Ple c~ eric ycr.t)
At r=R location
To determine porosity and permeability the core'spressure response has to be matched with Eq.14.All parameters of Eq.14 can be calculated exceptporosity and permeability. In terms of these twovariables this equation is nonlinear. The determination of nonlinear parameters is usually tediousand needs certain mathematical methods as iteration, gradient search or the Newton-Raphsonprocedure. To realize them is in general complicated, so a simple, but stable direct search algorithm was constructed. It minimizes the sumof squares between the measured and calculatedpressures. Fig.5. shows the procedure. Startingwith an initial guess for permeability and porositya rectangle is constructed and for all cornerpointsand side midle points the sum of squares is calculated. This means a search roughly in all possibledirections. One of the points will have a minimum function value and the rectangle is shiftedto this point. Again the function is calculatedfor the new points exept for those which overlapwith the last rectangle. If the function minimumis found to be in the midle of the rectangle, thesize of it is shrinked and the search goas on untilthe rectangle size becomes less as a certain limit.The method has the advantage that no derivatesof the least squares function have to be calculated.Also it is rather insensitive to the initial guessesand not susceptible to find local minimums on thesum of squares function surface.
Measurements have been carried out on four coresamples. The flow medium was formation water. In order to test reproducibility of the methodmeasurements were conducted several times. Therepetition of measurement gave nearly identicalresults, so the procedure has good reproducibility. The matched in- and outlet pressures of theparticular cores can be seen in Figs. 6-9. Results are summarized in Table 2. Also the resultswere compared with conventionally measured permeability and porosity values. As can be seen inFig.10 the agreement is good.
(24)
(21)
(20)
(23)
(22)
Ml (r1D ) = CIPI { J l (a..R)Yo(a..r1D )
Yl (a..R)Jo(a,.r1D )+
+c:; a,. [ Jo(a..R)Yo(a..r1D )-
Yo(a..R)Jo(a..r1D ) ]}
M2(r1D ) = 2C2P2'll'a..r 1D
Ml(R) = C2P2 { Jl (a..r1D )Yo(a..R)
Y l (a..r1D )Jo(a..R)+
c:; a .. [ Jo(a..R)Yo(a..r1D )-
Yo(a..R)Jo(a,.r1D ) ]}
M 2 (R) = 2CIPl'll'a..R
C = I-£tPc
Ie
whereas if r=R, then
In equations 14-23 C means:
308
SPE 22688 A.GILICZ 5
pressureP - Pini
see Eq.A-26radiusouter radius of coreLaplace variabletimevolumevariablesecond kind, zero order Bessel functionsecond kind, first order Bessel functionsecond kind, second orderBessel functiongas deviation factor or .;asroot of transcendent equationviscosityporosity
=
==
=
=
=
=
=
st
V
za
£.M
Npp
Q
n
r
R
NOMENCLATURE
A = constant in Eq.A-IO, A-llB constant in Eq.A-IO, A-llc = compressibilityC = see Eq.24C1 see Eq.lOC2 = see Eq.13F1;2 = seeEq.17,18Fa see Eq.B-6.H = thickness of core1 = imaginary unit or initial10 = modified, first kind, zero order
Bessel functionmodified, first kind, first orderBessel functionmodified, first kind, second orderBessel function
= first kind, zero order Bessel functionfirst kind, first order Bessel function
= first kind, second order Bessel function= permeability
modified, second kind, zero orderBessel functionmodified, second kind, first orderBessel function
K 2 = modified, second kind, second orderBessel function
= Laplace transformation signnumeratorindex
= denominator- The analytical solution is general, no ne
glections on pore volume storage were done.Also it includes particular cases where theratio of upstream/downstream volumes iseither much higher or much lower than 1.
- A simple but stable history matching algorithm was designed to determine porosityand permeability. So the evaluation is freefrom subjective judgement.
The results obtained by the method arein agreement with conventionally obtainedporosity and permeability values.
- As the method is fast it could be introducedas a standard measurement procedure.
- The lab apparatus allows measurements under reservoir conditions (confining pressure,pore pressure and temperatures).
CONCLUSIONS
As pointed out by Haskett et al. [8] the simultaneous determination of porosity and permeabilityby history matching has several advantages, sothe core has to be mounted only once, evaluationof results is free from subjective interpretation,stress dependent hysteresis is eliminated.
The method is faster and easier to perform as conventional steady-state methods, so as a new alternative it could be introduced as a standard labmeasurement.
The method fits well into operations made byradial models and does not increase operationalcosts.
The application of the pulse decay method hasbeen extended from linear to radal cores. Radialcores need a somewhat simpler preparatory workas linear cores (plugs) and may more properly represent flow conditions near the wellbore.
Unfortunately our apparatus could not captureearly time data with adequate resolution, so theinvestigations for core heterogenities could not beperformed. Future developments planned on theapparatus will make it possible.
DISCUSSION
Analytical solution could be found of thepulse decay technique for radial cores.
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6 APPLICATION OF THE PULSE DECAY TECHNIQUE TO RADIAL CORES SPE 22688
8. Haskett, S.E., Nahara, G.M. and Holditch,S.A.: "A Method for the Simultaneous Determination of Permeability and Porosity inLow-Permeability Cores" SPE 15379 paper,SPE Annual Technical Conference and Exhibition 1986, New Orleans, October 5-8.
SUPERSCRIPTS
- = Laplace transformed or average
ACKNOWLEDGEMENT
The support and permission of the Hungarian Hydrocarbon Institute to publish this paper is highlyappreciated. Special thanks to Dr. Gyula Milleyfor supervising the lab measurements.
REFERENCES
9. Kamath, J., R.E. Boyer and F.M. Nakagawa:. "Characterisation of Core ScaleHeterogenities Using Laboratory PressureTransients" SPE 20575 paper, SPE 65thAnnual Technical Conference and Exhibition, New Orleans, September 23-26 1990.
10. R.A1-Hussainy,H.J. RameyJr., P.B. Crawford: "The Flow of Real Gases ThroughPorous Media" JPT (May 1966) 624-636.
1. Brace, W.F., Walsh J.B., Frangos,W.T.:"Permeability of Granite under HighPressure" Journal of Geophysical Research(1968), Vol.73, 2225-2236.
2. Trimmer, D.A.: "Design Criteria for Laboratory Measurement of Low Permeability Rocks" Geophysical Research Letters(1981), Vol.8. No.9. 973-975.
11. Meunier, D., Kabir, C.S. and Wittmann,M.J.: "Gas Well Test Analysis: Use of Normalized Pressure and Time Functions" SPEFormation Evaluation, (Dec.1987) 629-636.
12. Van Everdingen, A.F. and Hurst, W.:"TheApplication of the Laplace Transformationto Flow Problems in Reservoirs" PetroleumTransactions, AIME, (Dec.1949) 305-324.
APPENDIX A
13. Abramovitz, M. and Stegun, I.A.: Hanbookof Mathematical Functions Dover Pub!.Inc., New York 1972.
14. Fodor, Gy.: Technical Application of theLaplace Transform (in Hungarian) Techn.Ed. Budapest, 1966.
The mathematical solution of the problem is similar to the solution of Van Everdingen and Hurst[12] except that the boundary conditions are different. Considering Eq.2. and applying Laplacetransform to Eq.1 and to initial and boundaryconditions i.e. to Eqs. (5)-(9), (11) and (12) weget
(A-2)
(A-I)tPP 1 dP -+--=CsPdr2 r dr
3. Lin, W.:"Parametric Analyses of the Transient Method of Measuring Permeability"Journal of Geophysical Research (1982),Vol.87, No.B2, 1055-1060.
4. Bourbie, T. and Walls, J.: "Pulse Decay Permeability, Analytical Solution andExperimental Test" SPEJ (October, 1982)719-721.
6. Amaefule, J.O. et al.: "Laboratory Determination of Effective Liquid Permeability inLow-Quality Resevoir Rocks by the PulseDecay Technique" SPE 15149 paper,SPECalifornia Regional Meeting, Oakland, 1986April 2-4.
5. Chen, T. and Stagg, P.W.: "Semilog Analysis of the Pulse-Decay Technique of Permeability Measurement" SPEJ (December,1984) 639-642.
7. Hsieh, P.A. et al.: "A Transient Laboratory Method for Determining the HydraulicProperties of Tight Rocks - I. Theory" Int.J. Rock Mech. Min. Sci. and Geomech.Abstr. (1981) Vol. 18245-252.
(A-3)
(A-4)
310
SPE 22688 A.GILIGZ 7
By introducing z == VCi the solution of Eq. A-Iis
dPb(r; a) = Azlt(zr) - BzK1(zr) (A-7)
The constans A and B have to be determined fromboundary conditions. Substituting Eq. A-6 andA-7 into Eq.A-3 and A-5 also considering A-2 andA-4: (A-13)
peri 8) = DIet (C1Pl {[ zIl(zR) + C28Io(ZR)]
Ko(zr) - [ C28Ko(zR) - ZK1(ZR)] Io(zr)}) +
D~t (C2P2 {[ zKl(zr",) + C18Ko(zr",)] Io(zr)+
[ zIl(zr",) - C18Io(zr,.,)] Ko(Zr)}) (A-12)
where
Det = [ zIl(zr",) - C18Io(zr",)]
[ C28Ko(zR) - ZK1(ZR)] +
[ zKl(zr",) + C18Ko(zr",)]
[ zIl (zR) + C28Io(ZR)]
(A-5)
(A-6)Per; a) = Alo(zr) +BKo(zr)
and it's space derivate:
For later use this solution can be put in the following simplified form:AzI1(zr",) - BzK1(zr",) = Cl{8 [AIo(zr",)+
BKo(zr",) ] - PI} (A-B) - M(a)P(r·a) =--
I N(a) (A-14)
AzIl(zR) - BzKl(zR) = -C2{8 [AIo(zR)+
BKo(zR) ] - P2} (A-9)
After rearranging:
A [ zIl(zr,.,) - C18Io(zr",)] - B [ zKl(zr",)
+C18Ko(zr,.,) ] = -C1Pl (A-IO)
M(s) and N(s) designating the numerator and thedenominator respectively.
One of the outstanding features of the Laplacetransform is that early and late time solutions canbe obtained without the complete inversion of theLaplace domain solution. This also allows the indirect check of the solution as both early and latetime behavoiur can be estimated by simple physicallaws. In our case we expect from early timesolution that it should be close to the initial conditions, whereas late time solution (steady state)should obey Boyle's law, as the process is actually isoterm expansion of a fluid through certainvolumes.
The early time solution in Laplace space is obtained by
A [ zIl(zR) + C28Io(ZR)] + B [ C28Ko(zR)-
zK1(zR) ] = C2P2 (A-ll)
P(rjt ~ 0) = £-1 {lim P(rja)}._00Invoking the equalties
(A-15)
Applying Cramer's rule this equation system canbe solved for A and B. After some rearrangementthe solution in Laplace space is: (A-16)
311
8 APPLICATION OF TITh JLSE DECAY TECHNIQUE TO RADIAL CORES SPE 22688
and applying A-15 we get after longer manipulations for r = r w
- PIP(r",; 6 -+ 00) = ( )
Vi Vi + JC/C~
and for r=R
- ~P(R; 6 -+ 00) = ---,---=----.-
Vi (Vi + Jc/c~)
(A-I7)
(A-I8)(A-2I)
Mter some manipulations
which can be inverted directly via tables [14] andwe get Eq. 25 and 26 in the text. They clearlytend to P1 and P2 respectively if time is close tozero, and these are actually initial conditions, i.e.Eqs. 6 and 7 in the text. These solutions also havesignificance in core heterogenity determinations aspointed out by Kamath et al.[9].
Next the steady state solution is investigated. InLaplace space it can be obtained by
IPoo = ---
Denom
( CIPI { - [.;c;~R + C26 ] In ( .;c;r)-
[C2 61n (.;c;R) +.;c;~R]}+
C2P2{[.;c;~r.. - CIS In (.;c;rw )] -
[va."';" -c••] In ( va.,)} ) (A-20)
Recalling Eq.2 this means in real pressure terms:
(A-22)
(A-23)
(A-24)
p. _ V"PI + VtlP2
00 - qnrH(R2) + Vel + V,.
As can be seen the inversion needs the derivationof the denominator (A-13) with respect to s, theLaplace variable and the calculation of it at polesSn = 1,2, ... ,00.
Let's put the denominator of (A-13) N(s), into aform
which is actually Boyle's law as expected. We see,that both early and late time parts of the generalsolution fulfill expectations indirectly validatingit.
It can be seen, that if all volumes are in the samerange, the solution is sensitive to the porosity,whereas if the vessel volumes are much larger,they dominate the steady state pressure theporosity having neglible effect.
To obtain the complete inversion of Eq.A-12 thedenominator Det (A-13) had to be investigated forbranch points and poles. It was proved that therewas no branch point at the origin in Eq.A-12, onlya simple pole and all other singularities lie on thenegative real axis in the Laplace domain. Becauseof this the expansion theorem of Heaviside couldbe applied directly [14]. The form of it is
(A-19)
1KO(:I: -+ 0) = -In:l:;K1 (:I: -+ 0) =-;
:I:
Poe = limsP(r;s).-+0
:I:10 (:1: -+ 0) =1;11 (:1: -+ 0) = -j
2
so Eq.A-12 becomes
Invoking the following equalities [13]:
where N(s) = z4Q [z(s)] (A-25)
312
SPE 22688 A.GILICZ 9
where
and after some manipulations we get Eq.19 in thetext:
(A-32)
Ko{i:l:) = - ~i [ JO{:I:) - iYO{:I:)] ;
K1{i:l:) = -i [J1{:I:) - iY1{:I:)] ;
(A-27)
(A-26)
dN dQdz- =4z3Q [z(a)] + z4__da dz da
Derivating (A-25) with respect to s we get
If we calculate this equation at the s singularities,the first term drops, because Q [z{s)] = O. Thepoles are determined based upon this condition.Further
dz C .dz Cz3
-= -;z - =-ds 2z ds 2
So A-27 finally becomes
dN C [ 3dQ]-(a =a..) = - z -da 2 dz ..=....
(A-28)
(A-29)
n = 1,2,3, ... ,00
So a .. are the positive roots of Eq.19 and can becalculated later on numerically. Secondly we haveto calculate !f2- at z = ia... Invoking the equalities:
d1o(az) I ( )dz =alaZj
First we have to calculate the s.. poles. Clearlythere is one pole at the origin in A-25. For theother poles lets designate arbitrarily
(A-33)
and after some rearrangement we get
lo{i:l:) = JO{:I:);
(A-34)
(A-30)
(A-31)
where a .. are certain real positive values. This settlement allows simplifications in later derivations.Using the deffinition of z we get
To find the values of a .. lets substitute A-31 intoA-26 and make it equal to zero. Invoking the following equalities
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10 APPLICATION OF THE PULSE DECAY TECHNIQUE TO -RADIAL CORES SPE 22688
Taking Eq. A-34 at Zn = ia" we get after longmanipulations
where Fl and F2 are Equations 17 and 18 in thetext.
Substituting Eq. A-30 and A-31 according to A24 into the first and second term of the numeratorof Eq.A-12 we get
M1(r) =
i a " {[J1(a"R) + c:; a"Jo(a"R)] Yo(a"r)-
[Y1(a"R) + c:; a"Yo(a"R)] Jo(a"r)} (A-36)
M 2 (r) =
ia" {[da"Yo(a"r..,) - Y1(a"r..,)] Jo(a"r)-
[d a"Jo(a"r..,) - J1(a"r..,)] Yo(a"r) } (A-37)
Finally substituting the last two equations alongwith Eq.A-35 into Eq.A-24 and after some rearrangement we get Eq.14 in the text. Note thatthe steady state term Poe in Eq.14. appears dueto the simple pole at the origin in the Laplacedomain.
APPENDIX B
As mentioned earlier, three possibilities exist considering volume ratios. The solution for all volumes beeing in the same range has been describedin Appendix A. Two other possibilities remain:Vu » VdjVu « Vd
Case Vu » Vd
The solution of this problem can be derivedfrom Eq.A-12 taking the limes of it if Vu --+
00 i.e. C1 --+ 00. Performing this and makingsome simplifications we get
z [Kl(ZR)IO(Zr) + Ko(Zr)I1(rR)]- PIP(r; 8) = -;- ---=-----,D=--en-om-----~+
8C2 [ Ko(zr)Io(zR) - KO(ZR)Io(Zr)]
Denom (B-1)where
Denom = z [ K 1(zR)Io(zrw) +KO(Zrw)Il(ZR)] +
8C2 [ Ko(zrw)Io(zR) - KO(ZR)Io(Zrw)]
Mter a similar treatment as in Appendix A theinverted solution is:
![Diodes Based on Pyreno[4,5-d]imidazole-Anthracene Highly ... · The transient EL decay was tested by an Agilent 8114A pulse generator (100 V/2 A) to generate rectangular pulse voltages.](https://static.documents.pub/doc/80x56/5f0ca1e97e708231d4365ec2/diodes-based-on-pyreno45-dimidazole-anthracene-highly-the-transient-el-decay.jpg)
