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Permeabi l i tya Mul t iprobe

Det erm inat ion Wit hFormat ion Test er

Peter A. Goode,; SPE, and U.K . MkhaeI Thambynayagam , q q SPE, Schlumberger-Doll Research

We aO-73f7Summary. This paper presents analytic models to interpret the pressure transients measured by a multiprobe formation tester. Themukipmbe tester discussed consists of three probes. The sink Probe generates a pressure pulse by withdrawing fluid from the formationwhile the resulting pressnre response is measured at the sink probe and at each of two observation probes. One observation probe ispxitioned on the opposite side of the borehole on the same vertical plane as the sink, and the other is displaced vertically on the sameazimuthal plane as the sink. The effect of the various reservoir parameters on the pressure response is discussed. Application of themodels is demonstrated through m aCNd field example.

In t rod t tc t fonThe introduction of the Repeat Formation Tester (RFTsM) tool inthe mid-1970s coincided with the exploitation of tbe North Sea. 1The RFT tool, which is primarily a device for messming verticalpressure distribution in open hole, soon gained industry recogni-tion. While the RPT technique has been successful, it has notreached its full poteritial because of its inability to give reliable per-meabili~ estimates. The estimated formation pammsters and thecollected fluid samples generally are from the contaminated zonerather than the native resemoiq consequently, the a.ssmiated degreeof uncertainty is high.With pmdcticm decline of many of the worlds large resewoirs,

reservoir management is receiving increasing attention as empha-sis shifts from primary to secondary operations. As secondary andtertiary schemes become tie dominant means of recovety, the mws-urement of permeabfity in heterogeneous and anisotropic forma-tions will become increasingly importsnt. Becsuse of its effect onmany reservoir displacement processes, vertical permeability oftenis the single most important reservoir parameter. Although verticalpermeability measurements routinely are made from cores, reliablein-situ measurements of vertical permeability over a large rockvolume are much more desirable.The transient well tests proposed for estimating vertical permeab-

ility may be ckmsitied as vertical imerference testimg. These tech-niques have not been widely accepted because they usually requirecased holes, and the measurements are masked by weUbore storageeffects. To eliminate these effects, longer (lence costly) testing timesare requimi For these reasons, industry needs an altcmslive meansof estimating large-scale, in-situ vertical permeab~hy.Moran and Finkled first proposed quantitative methods for es-

timating permeability using pressure-transient &w obtained fromtie wireline formation tester, the predecessor to tbe RFT tool. 3Their methcd corresponds to a tinitc, spherically shaped sink ofradius rp in an intinite, isotropic mti]um. During drawdown, thespace around tie perforation can be 61vided mugbly into thee spher-ical region% a steady-state zone near the sink where the total flowrate is independent of distance r from the sink, the undisturbed for-mation fa from the sink where the flow rate vanishes, and the tran-sition zone between the two where the flow rate decreasescontinuously from tie constant steady-state vslue to zero at somedistance far from the sink. During drawdown, as time increases,the boundmies between successive zones move farther from the sink.For sufficiently targe times, the Moran and FinMea spherical solu-tion yields a steady-state pressure difference,Ap,s=qfii4~krp. (I)The actual geometry at the RFT probe is not of a spherical source

but of a disk source. Therefore, an effective probe radius, r,, mustbe defined. For an isotropic medium, r, was defined as one-halfthe probe radius. 4.5 Shanna and Dussan6 later showed that thisdefinition was not valid and that the correct result is re =2rP/u.

Now at Schlumbqm Ow,eas S-A.-. Now at Schlumtarw Well Sewicm

Copyr ight $992 Soci@ty of Pelrdeum Engineers

The RFT probe enters the formation fmm the borehole. Becausethe borehole wall is considered impermeable, tie flow pattern cannothe spherical. At the sink probe, in the liit of steady-state flow,Stewart and W1ttman5 accounted for this wellbore effect in termsof a shape factor by writing Eq. 1 in the form

Ap, , =C@12~kre). . . . . . . . . . . .. .. . . . . . . . . . . . . . .. . . . . . ..{2.The shape factor, C, is essentially one-half the ratio between thepressure drop at a specified point with a borehole to the pressuredrop without a borehole. Depending on the borehole dmeter, Cvaries between 0.5 for spherical flow and 1.0 for hemisphericalflow. For an 8-in. borehole in an isotropic formation, Stewart andWittrmn defied c as 0..545 when r, was taken as one-half theprobe aperture. radius.Wifkinson and Hammond7 developed analytic~ derivations for

calculation of steady-state shape factors as a hmction of rp fr~ andanisotropy. They found that, for isotropic formations with a smallpmbe-~wellbme radius, the effect of the wellbore was small. Whenrp /rW =0.05, the effective shape factor was found to be 0.96 (itwould be unity for an infinite-radius well) when the correct detiti-tion of tbe effective probe radius r, =2rJ7T ws wed Fmanisotropic formations with k, < 4. Because tbe Reynolds number is proportionalto the fluid veloci~ and, in spherical flow, the velocity decreasesas W, tie non-Daccy flow phenomenon is localized to the sinkprobe only and bas negligible effect on the observation probes. Thispaper emphasizes perrmabifity derived from the presure &tarecorded away from the sink probe.The multipmbc concept originally was proposed in a U.S.

patents in 1956. Ffg. 1 shows the probe geometry of the mul-tiprobe td discussed here. fn operation, one probe (the sink) with-draws fluid tlom the formation. The resulting pressure transientis measured sinudtanmusly at the sink and the observation probes.From tiese pressure data, both vertical and horizontal permeabili-ties can be determined. Also, the observation probes am not af-fected (at least, not to Ieadmg order) by the adverse flow effectsat the sink probe. Because of the dktance between the observationprobes and the sink probe, the permeabithies are representative ofa larger length scsfe than those determined by a single-probe tool.

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horizontalbservation probe

VerticaJobservation prolx

Sink probe

Fig. lSchematic of a mulfiprobe formation tester. ~rlk.

Mathemat ic a l ModelThe mukiprobe geometry is modeled by considefmg the sink probeas a point sink on tbe side of an impermeable cylinder (the well-bore) in an anisotropic medknn, extending infinitely in the radialdirection, and unbounded vertically. A slightly compressible fluidis withdrawn through the sink at a constant rate. Gravity effectsare neglected, and the prccess is assumed to be isothermal. Modelingthe sink probe as a point source is satisfactory because tie obser-vation probes are dispkicd far enough from the sink that the actualgeometric details of tie sink are unimportant.The following equation (developed in Appendix A9) describes

the pressure chmge (in consistent units) at (rW,O,z) caused by apoint source of strength q located at (rW,O,O):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(4)

and~=k,r/@czr~. . . . . . . . . . . . . . . . . . . . . ...(5)Because we are interested in evaluating Eq. 3 only at tie observa-tion probes, it may be simplified to

p j~qm,p)q(~-e) .: . . . . (6)HP r)= ~T1.5~rW ~ /31.5

for the horizontal probe positioned at (rW, !r,O) and

. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . ...(7)

20

Vertical Probe1.6 -

~ 1.2a-Ya

0.8 -

0.4 - Horizontal Probe(6= 70

0.:04 , , I~o-2 10 102

Fig. ZPlot of G(OJ?) vs. D for 0=0 and e= T.

An alternative mathematical model where the mobe is ccmsid-... . . . ..xl as a i%ite-radius disk is presented in Appendx B. 9 Generally,

,.wever, the details of the sink probe are irrelevant for computingthe observation probe pressures, and the mmputational advantagesof Eq. 3 make it the most suitable model. Tbe function G@,L3),which incorporates the effect of the wellbore (Fig. 2), need onlybe calctiated at 0=0 (the sink and vertical observation probes) andO=r (the horizontal observation probe); it is not a timction of anysystem parameter. It therefore can be generated once and tabulatedas a function of O for each value of O and table lookup used duringtke evahation of Eq. 3. A cmnputatiomlly efficient expression isimportant if the model is to be used in an inverse solver.Figs. 3 and 4 show the pressure response at the horizontal and

verticzl probes, respectively, in response to a withdrawal of 10cm31sec for an unbounded formatim h&ing tbe properties tabu-lated in Table 1. Fig. 3 shows that the transient period at thehorizontal probe is relatively short and that the pressure approachesa steady-state value with @HP,, z U(krkz) ~. This is in keepingwith the results developed in Appendix C9 for a point source inznisotiopic media.The vertical probe reponse (Fig. 4) shows a much longer tran-

sient behavior than the horizontal probe, which is consistent witithis probes greater distance from the sink. The pressure at thisprobe also is asymptoting toward a steady-state value with@vP,= W It is s0mewh2t COU*W tO 10#c tit tie steady-s~~pressure at the vertical probe should be independent of the vefiicalpermeability, but as shown in Appendix C, 9 it is a correct result.ln t .arpret at ion of Mult ip robe Result sTraosient Interpretation. For the purpose of developing an in-terpretation scheme only, the case of an unbounded (vertically)reservoir producing at a constant rate is considered. Also, pres-sure data recorded at the sink probe are not used because of theirunreliability (discussed previously).There arc two approaches to developing an inverse procedure

with the data from observation probes. First, an inversion proce-dure that uses the data from both probes simultaneously could beused. Second, the responses could be treated separately, as thoughthey are not coupled. To determine which parameters each probeis sensitive to, we investigate the second approach. AISO, if the ver-tical probe is hydratdically isolated from the sink (as may often bethe case), a method that uses data from both the observation probes

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30 } , I , I

25 -

20 -

15 m _

L=.lll10 . k

5 L=lk,

/,~o 20 40 60 80 100

time, sec

Fig. 3Horizontal probe response uslngEq. 6 and the rockand fluid properties from Table 1.

20- , 8 , ,

L6 - f= I.- ,

z.2 12 -2gQ~ 0.8 -3:&

0.4

0.00 20 40 60 80 100

time, sec

Fig. 4Verficai prdbe response using Eq. 7 and the rock andfluid properties from Table 1.

It is shown in Appendix D9 timt, from the transient-pressure &iarworded at the horizontal observation probe, the quantities k,l+pc,and (k,kz) ~ [P can be determined, and flOM the vertid ObSe~-tion probe, we obtain kz/@c, and k,/P. fherefofe, assu~g hatthe formation is relatively homogeneous, it is possible to determinek, /p, kzlfl, and @c, with the mukiprobe geOMefV. lf tie vefli~probe were hydraulically isolated from the sink (e.g., a shale lay-er), then kz and k, still could be determined from the horizOnt~probe data if the prcduct of porosity and compressibility, .$ct, wasindependently known.By comparing the quantities measured at each of the probes, we

see some redundancy in the measured quantities. This redundancycan be used as a measure of the quality of the inversion. Unsatis-factory results most likely would result from the obscwation probesbeing influenced by regions of the formation having different prop-erties (i.e., the formation is not homogeneous). If the fonnztionis not homogeneous, more complicated mcdels are required.

Steady-State. If the rate of fluid withdrawal remains constantand the reservoir is unbounded, a steady-state condition will be aPpreached. The steady-state pressures can be used to deduce the rese-rvoir parameters if the resewoir is relatively uniform. .We begin to develop expressions for the steady-state pressure

by considering the expression that describes the pressure drop causedby a continuous point sink at (rW,O,O) (see Appendix Cg):

Ap(t) = q [mlzti4+ct&

m. (8)

If r=rw, @=0, and z >0 (vertical observation probe), Eq. 8 sim-plifies to

()pvp(t)=xerfc L . . . .. . . . . . . (9 )

and when r=rw, z =0, and 8=T (horizontal observation probe)w ( )wAPHP (f]= 86rw dc G . . . . . .

Using the series expansion 10 yield32 m (l)w zn+l

erf(u) = E G .=0 n!(ti+l)

(10)

Therefore, erfc(u) may be written for u*O aserfc(u)= l-(2/fi)u+0(u3). . . . . . . . . . . .. . . . . . . . . . . .. (11)

This asymptotic approximation is accurate to 1% when the argu-ment u< 0.17, which requiresZ>8.65z2@cr/kz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(12)

for Eq.9 andt>34.6r~@c,lkr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(13)

for Eq. 10. Tbus, a plot of Ap vs. llfiyields a linear relationshipwhen r satisfies these inequalities.With the approximation described by Eq. 11, the pressure ditkx-

emce at tbe vertical probe is

(1 1

Pi-PVp(f)=s 4mkr Zvp ~ )

. (14)

At the horizontal probe, it is

(P 1 2 )PiPHp(o = 8- ,. ~ (15)After subtracting E.+ 16 from Eq. 17,

pVP@-pH/ I@)= Z%(;-:*) (16)

Even though the pressure difference at the vertical and horizontalprobes may exhibit a transient behavior, Lbe difference between tiemis independent of time. Fig. 5 is a plot of the pressure at the two

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TABLE IFLUID AND ROCK PROPERTIES

k,. md 100Zbt.r cm 70rw, cm 10q, cm Isec 10c,, psi1 2x1o-5/ L, Cp 1+ 0.2r., cm 0.556

observation probes with the parameters presented in Table 1 andkrlkz = 10.0. Note hat the pressure diference between the probesbecomes steady while the pressure remains transient at each of theprobes.Eqs. 8 through 16 apply to a point source and do not account

for the presence of the wellbore. The weUbore can be accountedfor by c&sidering a steady-state factor, The steady-state shape fac-tor, 0, is determined (Appendix E9) by comparing Eq. 8 with Eq.3 and is given by

where ~=(z/rW)m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (18)Ffg. 6 is a plot of the shape factors for 6 =0 and 8=T as a func-

tion of g. The values also are tabulated in Table 2. As expected,the shape factor approaches two as the displacement from the sinkapproaches zero (i.e., the wellbore appears to be infinite) aod unityastbe displacement approaches intinity, wheretbe welfbore hasnegligible effect. It maybe observed that the shape factor goes be-low unity for some range of ~. This is not a numerical ardfacGit is from wellbore distortion of the steandims, causing less fluidto flow along the wellbore and resulting in less pressure drop thanwould be expected if the wellbore were not there.The shape factors can be used to mcdify Eq. 16 to account for

the effect of the weUborm -.Pvp(o-%p(o= Z&(%-:d:) (9)

10

6

6

4

2

0

. . Horizontal probe/

. .

\

\Difference

,0.0 0.2 0.4 0.6 0.8 1.0

Spherical Time, sec*5

Fig. 5Plot ofp vs. IAIT for the k,lkz = 10 case presentedIn Figs. 3 and 4.

TABLE 2STEADY-STATE SHAPE FACTORS

zlrwm 6.0 R.T .zlrwm 9.0 e.u0.0001 1 .999s 0.s117 1.8411 0.5121

0.08000.0900

1.8765 0.51181.8642 0.51191.8524 0.5119

60.000090.0000

0.99s30.99s5

From Table 2, we can see that for kr/kz =1, Qrp =1.0203 (forZVP 70 cm and rW=10 cm) and for k,/kz= 10 , OYP =0.9974.Therefore, in Eq. 19 we can let OVP = 1 without intmducmg sig-nifkant error. Clearly, the weUbore has negligible effect on thepressure measured at the vertical probe. Additionally, f3Hp is nota function of tbe anisotropy because g = O. Therefore,f?Ho =0.5117, and Eq. 19 maybe written as

qfi (0.5117 1--J) &Pvpw-mfpw - . . ..(20)4= 2,. z Vp %20

1.6 -

s;

1.2 -L~m~~ 0.8::

0.4 -

0.0 , t t ! # t104 ~o.z 100 102

c

59. 6-Steady-state shape factors as a function of the scaledter tlca l d isplacement.

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Thickness = 100 cm-d 25 -z~.z 20 -$~Q

15 -Es% 10~ . . . . . . . . ----b . . . . . . . . . . . . . . -.Thickness=

5

. ! ,o 20 40 60 80 100,

Time, sec

Fig. 7Effect of vertical boundaries on the horizontal obser-vation probe.

fn addition to the difference in pressure measured at each of tieobserwmion probes, the pressure at each probe wilf asymptote toa steady-state value. For large time, F.@. 14 and 15 may be ap-proximated to (introducing the steady-state shape factors)

9PIim [pjp#)]=- .,, . . . . . . . . . . . . . . . . . . ..(21)l-a 4~k,zV

o.5117q/land Iim [P~PP (t)] = ~ ,., . . . . .,-. m rzlv

. .. . . . . ..(22)

respectively. By combining Eqs. 21 and 22, we can determinekH/# and kv/A if the reservoir is homogeneous. Additionally, tknnthe slope of the strzight-line portion of the plot of p (t) vs. 116dc, ah can be determined. Because the time taken to reach steadystate is a function of the distance between the sink and the obsema-tion point and the diffusivity in the direction of the displacement,the horizontal probe probably will approach steady state faster thanthe vertical probe. It maybe nec~smy to extrapolate the plot ofAp(t) vs. lf~to l/~=0 (infinite time) to determine the steady-#ate pressure at the observation probes, as illustrated by the dashedline in Fig. 5.

Use of steady-state analysis assumes that the flow rate remainsconstant and that the diffusivity ii sufficient for the approximationgiven in Eq. 11. If this were not the case, methods to deconvolvethe effects of the variable flow rate would be needed. 11 Also, ifvertical boundaries influence the pressure response, there will notbe a steady-state pressure at any of the probes. However, if a spher-ical flow regime is established around the probes before the influ-ence of the boundaries, then E@. 20 through 22 may still workby use of the extrapolated pressure from a plot ofp (t) vs. UfiIf the boundaries are too cbze to allow a spherical flow regimeto develop, the steady-state analysis cannot be used.Effec t of Vert ica l Boundar iesIf vertical flow barriers exist, the model presented by E@ 3 willnot apply at long times after the press% pulse has reached theboundaries. Therefore, the model can be extended to include thevertical boundaries with the images method. If the boundaries areat z=O and z=h, then Eq. 3 becomes (Appendix A9).

Thickne5s.200 an7hickness= zoom

//

--- -f-------7---- Thickness= .=0..-0.4

I/0,0 ~

o 20 40 60 80 100Time, sec

:Ig. 8Effect of vertical boundaries on the vertical obsewa-Ion probe.

Ap (t)=

, . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(23)

[z(2hizJ]2 k, [z(2hi+zs)12 k,where T,= ~, ?2= ;,. . (24)4f?r; , 4or; ~

and the sink probe is situated at (rW,O,zS).Figs. 7 snd 8 are plots of the pressure determined with Eq. 23

for the horizontal and vertical probes, respectively, for severalwdues of the layer thickness h, assuming z~=h12. To generatethese figures, the parameters tabulated in Table 1 and an anisotropyratio of k,lk= = 10 were used. Figs. 7 and 8 show that, if the thick-ness of the layer is sufficient (for a given due of kz), the pres-sure respopse is not affected by tie presence of the boundaries.This implies that the interpretation schemes dMcussed could be usedwhen tie layer issufficiently thick. However, 2s the scaled layertbichess decreases, this is no Ionger the case. If the test is runsufficiently long, radial tlow will develop and k , ILL can be deter -mined from the semilog dope, similzr to a well test.The transient interpretation scheme can be extended to includethe presence of vedical flow barriers. However, solving an inverse

problem containing the boundaries intscduces two additional pa-rameters h and .ts, the formation thickness and tie probe positionrelative to the boundaries, respectively. fbis makes the inversionless well-posed. If, however, h ardor zs were determined izzdcpen-dentiy from other wimfine logs, the resulting inversion would ksimilar in complexity to that of an unbounded reservoir.Field ExampleTh e data considered in this example were acquired tYom a devel-opment well in a large oil field that has been under pmducdon forabout 35 years. The formation of particular interest extends fromribout 410 m 440 ft and is divided into two zones by a shale at 426

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X400

X450ESP------ -So.o 20GR0.0 2W.. .,- E=a4Fig. 9Openhole logs for example.

30

40

30.$

*20

10

0

100

80

20

00 30 60 90 12 0 15 0

Fig. 10 Obsewafion probe pressurs responses and calcu-lated flow rate for test at 433 ft.

ft (Fig. 9), Both zones comprise homogeneous sands of high per- sink probe relative to this boundary was fixed in accord with themeabilitv. The field has undereone waterfloodine for some time Wsition Of the tool, while the location of the toD boundarv was a&and is b~ing pumped. Two multiprobe tests were performed with the multiprobe mod-

ule of the Modular Formation Dynamics Tester12 (MDTsM) at432 and 433 ft. Fig. 10 is a graph of the pressure responses meas-ured at the monitor probes for the test at 433 ft and shows the esti-mated flow rate. 13 The irregular pressure response at around 35semnds probably was caused by small volume changes in the teles-copic flowliie comecting the probe to the pressure gaugq thesearose from a hydraulic event initiated by the tool to maintain a sealbetween the probe pads and the side of the borehoie.A 2.evenberg-Marquardt 14 Parameter estimation procedure using

the bmmded, homogeneous, single-layer mc.iel described in F+29 was used to obtain best estimates for the radial and vertical no-bilities and @cr For tie initial stages of the analysis, the positionof the bottom boundary was freed at 439 ti. The distance of the

I I I , , , I

0. 0

0 10 20 30 40Horizontal AP, psi

lg. 1 IComparison of measursd and reconstructed pres-;ure responses for the test at 433 ft.

~usted within the r~ge indicated by the static p;essure prkks andthe logs (429 to 431.5 ft). The most consistent resuft was obtainedwith the top boundary at 430 ft. The results obtained by the pa-rameter estimation procedure were krlp= 129 mdlcp, kzlfi= 13.6mdlcp, and c$c, =3.8 x 10-6 psi-l. To visuafize the success of theinversion, the recomtructed prwsure response, constructed withthe estimated pamzneters, was compared with the measured data(Fig. 11). This plot style for the presentation of MDT muhiprobedata fust was introduced in Ref. 12 where properties of the plotare discussed briefly. The comparison in Fig. 11 of the measuredand reconstructed pressures has content because the parameter es-timation was performed with raze&corwoIve.d pressures. 1I There-fore, goodness of fit becomes a measure of both the values of thederived parameters and the adequacy of the deconvolution.

ConclusionsA new-generation formation tester with multiple probes has beenintroduced, and analytic equations are presented to model the toolr~mc in both vefidly bounded and unbounded reSeITJ&S. Themuhprobe configuration offers many advantages in terms of per-meabficj measurement over single-probe testers. Measurementscan be made away from the sink probe where adverse flow effectscan lead to quantitatively questionable results. The three-probe ge-oznemy of the MDT tool coupled with its enhanced flow capabili-ties permits investigation of much larger volumes of the formation.A3so, the presence of the vertical probe duectly determines the ex-istence of hydraulic mndnui@ between it and the sink.It was demonstrated that a multipmbe formation tester can de-

termine the horizontal and vertical permeabihties and the forma-tion storativiiy. Applicatiori of the models for interpretation of actualfield data taken by the MDT td was presented.

Nomencla tureA,B = arbitrary Comtsnts

c, = total compressibfity, atm 1 pa-1]C = Stewart and Wittman shape factorG = defined by Eq. 4h = reser.wir thickness, cm [m]

10 = zerc-order modified Bessel function of the first kindJo = zero-order Bessel function of the first kindJ1 = fret-order Bessel function of the fist kindJ;, = derivative of ntb order, Bessel function of the first kindk = permeability, darcies

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p = pressure, atm [Pa]~ = volmetric tkid withdrawl rate, ~3/,r = radial coordinate, cm

rw = wdlbme radius, cmt = time, secondsu = dummy argument for mafbematicaJ functions

Y. = zero-order Bessel fonction of tbe second kindY, = first-order Bessel function of the second kindY; = derivative of ntb order, Bessel function of the

second kindz = verticaf coordinate= tmJh, cmA = difference

-f I ,72 = defind by J%. 24q = diffusivity =kl+pct, cm2/s6 = coordinate radiansY = viscosity, q p.s]t = ZlrwwT = detined by Eq. 54 = porosiiy, fractionQ = steady-state shape factor

Subscripts~ = ~ffetiiye probe radius

Hp = horizontal observation probei = initial

p = actual probe radiusr = coordiitess = steady ststeS= sink

Vp = vertical observation probez = coordinate

Reference1. Schultz, A. L., Bell, W.T., and Urbmos!g, H.J.: Advancements in

UncAsed -HoIe, Wir eline Formation-Teste r Tedmiques, JPT (Nov.1975) 1131-36.

2 . Moran , I.H. and Finkka, E. E., Them-&d AI@sisof Fmsmre Pk .nomenoa Associated With the WireliIe Form a t ion Tes t e r ,, - JPT (Aug.1962) S99-908; Trans., AlME, 225.

3. Lebmrg, M., Fields, R, Q., and Doh, C. A.: Medmd of FormationTesting m Logging Cable, JJT(Sept. 1957) 260-6Z Trans., AJME,210 .

4. M.s k at, M,: Phyical Principles of Oi l Producdm, McGraw-Hi BwkCo. Jnc., New York City (1949).

5. Stewart, G, and Witbnan, M.: Ymmpretadon of the F7essure Respn.sesMfbe Rep-at Formadm Tester,,> p+m SPE 8362 pmenfed af fbe 1979SPE kmml Tecbn&d Conference and Exhibition, Las Vegas, Sept.23-26.

6. Sbarma, Y. and Dussa$ E. B.: Anafysis of fbe Pressure R.sPonse ofa Single-Prob. FormatIo@. Tester, SPEFE (June 1992) 151-56.

7. Wilkinson, D,J. and Hammond, P., LA P.murbatirm Theorem for MixedBoundary Value Problems in Pressure Tmmimt Tesfiog,, Trmponin Porom Media (1993) 5, 609-36.

8. Doll, H.G.: %fetbcd and AI&IMJS for Determining Hydraulic Char-acteristics of Formations Traversed by a Borehole, U.S. Patent No.2,747,401 (1956).

l Peter A. Goode isthe chief reservoirengineer for Sch.Iumberger in Southand East Asla. We.viously, he workedas a research SCI.enfkl at .schlumber.n&.,-.,:+

e.* :- i

ger Doll Researchin Rldgefield, CT,and in various

Goode Thambynayagam reservoir engineer.lng positions for

Sohlo Petroleum Co. and Santos Ltd. He holds a SS degreeIn mathematics from the U. of Adelaide and a PhD degree inpet foleum engineering from Hetfot-Watf U. Goode seties onthe 1992-93 Asia Pacific Oil & Gas Publications Committee.Mlc had Thambymww fam Is a manager of the lnterpreta-tiofl Engineering Dept. at Schlumbefger Well SewIces inHouston, TX. Prsvlou81y, he worked at Schlumbefger Doll Rs.search in R[dgefield, CT. He was manager of Reswvoir Studiesfor British Petroleum in San Francisco and a consultant forthe Computer Aided Design Center, an establishment of theDept. of Industry in England. He Io[ned Schlumbwger in 1985.Thambynayagam holds a PhD in Chemical Engineering fromthe U. of Manchester, England.

9 . Gc w ie , P.A. and Tbmbynayagti, KM.: Supplement to SPE 20737,Permeability Demmimdm With a MuOipmLE Fonmdm Tester,>, papwSPE 25396 .wai]able from SPE Bmk Order Depr., Richardson, TX.

10. Abmnowitz, M. and Stegun, I.A.: Handbook of MatheMcal Fmc-tiom, Dover Publications Jnc., New York City (1972) 297.

1 L Kuchuk, F. J., Carter, R. G., and Ayesbman, L.: Deconvoludon ofWelfbore Pressure and Flow P&% SPEPE (March 1990) 53-59.

12. Zimmenmn, T. eral.: Application of Emerging Wirdine F.ommtionTesdng Technologies, paper OSEA 90105 presented at fbe 1990 Off-shore South East Asian Conference and Exhibition, Singapore (Dec.4-s).

13, Gocde, P. A., Pop, 1.3., and Mwphy, W.F. JJL Muldple-Probe For-mation Testing and Vertical ReswvoiI Comin.ily, y paper SPE 22738P~eo~ atthe991 SPE Annual Technical c,onfmerI~ and Exhib,.non, Dallas, Oct. d-9.

14. Marqumdt, D.W.: Algorithm for Lesst-Squares Estimation on Non-Jinesr Parameter , J. .SOC. lnd. APPL Mark. (June 1953) 11,431-41.

SI Met r ic Conversion Fac t orsq x 1.0 E03 = Pasft x 3.04S* Eol = m

in. X 2.54* E+IN = cmmd x 9.869233 E04 = pmz

psi-l X 1.450377 E01 = kpa-l.C4nvmslon fac!or is .9x-L SPEFEoriginal SPE manuwrbt m.dved for review Sept 2, ?990. Revised mmuscripl receivedFeb. 10,1992. PaFwaccqleC IorDbllcation APtil 7, 1S92, Paw (SPE 2)737) firs! Pr6ssnt-ed 81 %he 1990 SPE Annual Tffihnia Conference and Exhibit 1.. held in NW Orleam,Se pt . 2 z- 26 .

SPE 2S?S6, 5uDpletnan1 to SPE 20737, Permeabimy Dwmlnat!.n Wiih A MultiPmbaForma!lo. Tester; wllame from SPE Bc.ak Order DeP!.

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8 Ansdytic Models fcmrXMultiple Probe Formation Tester SPE 2

I

12.

13.

14.

15.

16.

17,

18.

19,

20.

21,

22,

23,

240

ing, %n Antonio, Texas SPi3 10209 (1981).Stewart, G,, \Vittmann, M., and Van Golf- Racht, T.: TheApplication of the Repeat Formation Tester to the Analysisof Naturally Fractured Reservoirs, Presented at the .5GthAnnual Fall hiecting, San Antonio, icxas SPi2 10181 (19.81),Stewart, G. and Wittman, h!.: Tl]e Interpretation of Dis-tributed Pressure and Flow hfeksurcrnents in ProducedReservoirs; Presented at EURGIEC, paper 272, Royal Lan-caster Hotel, London, L. 1{, (1S82).Stewart, G. and Ayestaran, L.: The Interpretation of Ver-tical Pressure Gradients Measured at Observation \Jr+ls inDeveloped Reservoirs, . .rc~p!,tcclt~7thannual Fall llect-ing., New (.)rleans, Louisiana (1982).Bilhartz, H, L., Jr.: J.?ffccfs of wellbore Damage and Stcrra~eon !3chavior of Part ial/y-]~(;l ]etr; l t ing \te/ls , Ph .D, Thesis ,Stanford [University Stanford, California (1973).Elurns, W. A., Jr: New Single-\Vell Test for DeterminingVertical Pcrmeability~ J . Pd. Tkh. (June 1969) 21, 743-752,Prats, hi.: A hlcthod for I)etcrmiuing the Net Vertical Per.mcability Near a \Vcll from in situ Measurements, J . Pd .TeclI, (May 1970) 22,637-643.Falade, G, K., and Brigham, \V. E,: The Dynamics of \~erti-cal Pulse Testing in a Slab Reservoir, Prcseuted at the WthAnnual Firll Mcwting, Houston, Tcxirs, SPE 5055-A (197.1),I%dadc, G, K., ancl Brigham, IV, E,: h Analysis ofSingle-\Vell Puisc Tests in a l~inite-Acting Slab Reservoir, l>rc-sentcd at the 4!]th Annual Pall h~octing$ Houston, Texas,SPE 5055-B (1974).Earlougher, R, C,, Jr: Atlalysis and Design for Vertical \\eHTesting, J, M, hch, (hlarch 19S0) 32, 505-+14,IIirasaki, G. J,: Pulse Test and other Early Transient Pres-suro Analysis for in situ Estimation of Vertical Perri lcahi li ty,Soco Pet, Eng, J, (M. 1974) 14, 0-4),Gillund, C, N., and Katna l, M, M,: incorporation of Ver-ticid Permeability Test Results in Vertical Aiiscihle Floodi)csigu und Opcratioa ,, J, Cdrt Pet, (Miircll -April ]984)23(2), 54--W,I%tnul, M, M,: Intcrfcrcmcc and Pulse Testing - A Rcviow,J, Pet, Tech, (Dcccmber 1983) W, 2257-2270,i3rcmer, IL iL,Wirlston, H,, rind Vcla, S,: A.mlytical Modcdfor Vertical Intcrforcncc Twrt.s Across Low Pcrmcul)ilityZoncs~ Prwscntcd wt the Wth Annual Technird Confcrwnceand Exhibition of the WE, San Francisco, California,SPE

1

25,

26.

2?.

2P.

29.

30.

31.

32.

33!

24a

35!

36!

37,Respoasc of a Single Proljo Formation Tester, lrrxmntc

I

11965 (1983).Lee, S, T., Chicn, hi. C. H., and Culham, IV. E,: \~e:Single-\Vcll Pulse Testing of a 1hrec-Layer Stcatificd Rvoir~ Prcscntwf at ttie 59th Annual Technical Confcreand Exhibition of the !jpl~ , Houston, Texas, SPE 1(1984),Kunzman, \V, J. and Earlotlgher, R, C,, Jr,: Field Acation of Verticrd Testing hif?ttlods with a Crtsc I1istoPrcs>titcd at the 55th Annual 1,c linical Cwii,v,. ,icc andhibiticm, I)allas, Texas, SPE 9458 (1980).T,&*,,al, ~[. ~j.; The EfTccts of \Vcllbo. Storage and Ski\~crticai Permeability Testirrg~ Presented at the 59t!] AnTechnical Confer-cnce and Exhibition of the SPE, HousTexas, SPi? 13250 (1984).lN]lig-Economides, C A,, and Ayoub, J. A . Vertical Ifcrcucc Testing Across A Low Permeability Zone, Presenat the Mlth Annuirl Technical Conference and Exhibitionthe SP13, Houston, Tex~s, SPE 13251 (19S4),hloran, J. H. and Finklea, E. E,: Tlworctical AnalysiPressure Pb-snomcnon Associated with the \Virclinc l~otion lcstcr~ J , Pet Tech, (August 1962) 14, S99-908.Lclsourg, M,, Fields, R. Q,, and DoI), C. A.: *AhlethoFormation Testing on Logging Cable., J . Pet, TCCII (Scptber 1957) 9, 260-267.11uskat, if: ~>hysica/ Principk s of ~i] production , hlc GIIill, Ncw \ork (1%19),Stewart, G., and \Vittman, M: Intcrpretittion of thesure Response ot the Repeat Formation iester, Preseat t llC 54t.11 Annual TcCll llir ill Confcreace and Px]i ibitiolt hc SPE, Las \cgas, Nevada, SPE 8362 (1 !)70),\Vilkinson, i.). J, and ll iit l] ]] lot~d, P,; A lerturlxition Trem for hiixcd Boundary Value Prolrlcms in Pmssurc Tsient. lcsting, Submitted to irilnsport in Porous hi(January 1987),Doll, 11, G,: Lfcthocl a ad Apptsrw I.US for Dctrvvniningdra ulic (X nrac kvistics of lbrmn tions lfiI\JCJS(!dby iiIId c,, U, S . iatcwt No, 2,747,401, (1%56),Porsytllc, G. IL, hialcolm, hi. A. and Mcrler, C, B,: cputer Methods for Affithematical Corl]pUtiltiOlls, PrcuIiall, Ncw Jersey (1977).Al>ramowitz, M, and Stcgun, 1, A,: HatJdbook ofhfilthomictd Functiom, Dover Publications, inc, New York (1 !)7Sharma., Y, and l)ussuII U, 11,: An Analysis of the lrcs

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SPE 20737 P, A, Goode and R. K, M. 1lmmbynayngam

the Annual 1echtrical Uonfcrence and ~xhil.ri tion of the SPE,Dallas, Texas, SP~ 16801 (106?),

38. Carslaw, H. S. and Jaeger, J, C,: Conduction of Heat rnSolids, Oxford Science Publicaticms, Oxford, L?,K. ( 1959).

39, Morse, P. M. and Feshbach, H,: Methods of TheoreticalPhysics, McGraw-Hill Book Company, New York (1953).

40, Tranter, C, J.: In te gra l Ik a ns forn s s in Mt. thcmaticcd Physics) Methuen and C(). LTD,, I.ondon (1966).

41, Stehfest, H.: Numsrical Invcrsioil of Laplace Transforms,fhnmrsrrications of the ACAf, (1970) 13, 47-49.

42. Durrant, A. J,ar/d Thambyn&yaganl, R., K. M,: \lellLoreHeat Transmission and Gcothermrd Production: A SimpleSolution Techniqu~~ SPE Reservoir Engineering (h~.wch,1986) 2, 148-162.

APPENDIX APoint Source BoundcrJ ?nternal]y bya CylinderIn this appendix th: general salution for the multiprolre con-figuration including vertical borrm.laries is devoiopcd. Carslawand Jaeger 3S provide the solution for a unit instantaneous pointsource at (!!, O, 0) rtt time t = O in the region bounded internallyby the cylinder r = a the radiation into the medium at r = abeing kept at zero as

1hisequation can bc rwwrittcn to cb-wcribc flow of wslightly cols-pressible fluid in a homogeneous twisotropic medium, wlwrc thesource is locwtcd at (rw, O, Z*) and as

To obtain the solution for a cent.rnuous source Eq. A2 is intorgmtcdwith rrwpcct to time Logive

where00G(O,/3)= $ ~ cos(?to)/ pcaz~-da (A4)ti=-m o a (J/, (a)~ -t- }p~ = q}l (

(((((

(llw cqliittion (dropping the subscript D for convcnivnce)

and the following initial and boundary conditions:P =Oatf=O +- r,o Und :, (M=Oasr-w 40,2 Und t, (13

;=oato=o,r + r,s tinrl t, (1)$$=oil t:= o, h +- r,o r ind /, (11

1,.. -

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10 Ancdytic Models for n Multiple Probe I?ornmtion Tester SPE. . .-.. ..and

~{{

o 0> Jf-fy:~;$f= o (:,+12) (111:])

matllcmatical]y dcscrilxw flow into a hole of finite radius Q 011the surface of an impcrmcfildc cyiindcr cf radius r = 1, boundecivertically at $ = O and : = /1.

kollowing tlw application of two finite Fouricv cosine triill8-forms40 (in O apd : ) and a I.aplacc transform Eq, 13Stxcornes

wherej=oasv-eo,

illld

(U14)

(1315)

(1316)

(1317)

ing ~q. 1317and equating it with Eq. 1116 D can be dctcrmitwd

/ \

Succcssivc application of the Fourier invomion formula gives thefollmviug

8WR.+ ~ ~ ~ Glilll(s,l)=1)111aNllrT2) 41110) ,?11:=11= 1(1119)

\vhcrv d,,,,,(%)) = 21(,,, (0!)so [/(,,, +1(0) + 1(,)1-1((7)]illld

1120)

LY21)

L,),,, (s , r) is invcrtcwl uumcricullv usiug tlw algori thm devby Stchfcst,41 Al) iilVt ic invers ions arc :i\wiliii)lo for &o[)( ,s.60,, (s, r) ,42 however no atlalytir invorsioll rxists for G,:}

illl(l00Go,,(t, r) =? J [{~~-[(lw )%l?]f } +-iT(J

Thrsc ful]ctions lias bccu tabulaud hy l)urrant and l

Calculation of GlfiII(S, v) (Eq, 1120) can Iw difficultvalues of nt . For tl]is reason a rwurrmco relationship IdcIYclopcd which only rcquirws tho colljp(ltatim] of li.~ ii[lsillg tl]c rcititicsnsllips:]o

1(,,, +., (u) = - Iit,t + lit,, - 1 ,;1nd _= 21(,,, (11)

1(;,,(u) A,,t+.l (u ) + [(,,,-1(U) it CmI IIc shown that

Iilll ( II) ,2-=~ Ill = 1,2,,,,(}111+ ,,,

Whw 2/11

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SPE 20737 P, A. (%.sode and R. X, M, Thnmbynnyngnm,.Letting f;c = l(Y = Kr 13q. Cl bccomcs

where a = r*+ ?; - 2r?rJCos(tl - O.), A corresponding form cmbe \vrittrm for rmcrvoir crrgincering problems as

Ap(t)=q(qillct)i

1 {[+pc, Q + (s ..q \

$@,) - . .. .... .... ,814rc(~-, 4t h% i,, ,11 ((:;{)LMining L!, A, ~ Ml d = d~ then for a continuous point source, whcrw fluid is withrhwn ata rate q fron] timed to time= t Ap(t ) can !): obt~itwd i~:yintograt. ing Eqo C3:

+Tx {-++-l}d (c )t

Substituting u = (t - ~)-~Ap(t) = q Im ){-$ [+++}4(J+

A property \\!orth some iit.tc)~t.ion is that for I:trgc time,Ilq, CS appro?clm a stca{!y-stut.c prmrurc drop in the vucinilyof the siuk, This is peculiar to spherical flow, The rmu!ting

Also, if r = vo md O = 00 thcll Eq, (76 IXW!OIIIW(C7)

NIQ ~tewly.titatowwurw at L point in the nmlium duo to acoatinucrus pointsource d titrcngtlt q on u cylindrical intcrnat

221

boundary of radius rw, through which no flux passes cancxprmscrl as the steady-state pressure (IUC to a continsrou.- ,source in an infinite medium mtsltiplicd by a shape fac tor waccounts for the prcscnc~ of tnc wellbore,

Tcrdetcrtninc the shailc factor wc neccl to cotnparc the stestatc solutions of the point source in an infinite mdium withwithout a wellbore.

1hc+itncnsionlcss stcacly-stats pressure at a point (ruI ,duc to a point sink &t ( VW, 0,0) can bc rlctcrrnincd from Ilq.ss

For Iargc values of f this is not a computtttionally convenfortn and wc will switclt to the Fourier rcprcscntation givenEq, 1319, lhking the limit a ~ O (point source) and rwplacittgl)ourior sum in x with a louricr integral (rcmrwing the boundain s ) this equation lmcomcs (for stcariy state)

To m~provc the convcrgcncc fc: the second tcrtn tho followiCI II ;@Cd, I ly making the substitution $ = nm ,

may !Ic \vritt.rN as

The uniforn] expansion for l i\t~P order:]o is 11.-J- 2 ,2t)l (~ ,- ,,:);\\IolilxJrc) l;q, 1)3 CaII Iw \\

~ COs(mo)x!llkll1{ Ii,,, (/~nl) 1() --w ll~-Fl}cO(l)[l

w+,>1 c~s(tllo)lfo (m{)+g {lo,c(l-c-f)+*} ~

1hv stuady.stsstc prwwurc for rr ctmtinuous point sourr~) = I,SD =0 and U =0 frrs]tl Eqi (!6 is

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12 ~\nalytic Morkds for a Multiple Probe Formation Zkwtcr SPE

therefore tlw shape factor O is0 = y~ ~ (D7)

where p~~~ is cnlculatcd using cir,hcr ltq, D1 or ltq, 1)2, Forthe Wrlicid observation plwtsc O = O and { > 0, and for thehorizontal OIW.!IWltiOllprol>c O = rr ulld f = O ,

TIIC rmulting steady.state sl]iij)~ factors arc tabulated in Tu-Mo 2 at)d plotted sw u functiol] of f in l~ig, U.

APPENDIX EEfrcctiv( .r. ., lw Radius (Siok J)rolm)11) rcidity tl]c sink prol)c should lx] Inocfclvcl iss ~ Imlc of ~iiditi~71)over which tllo pressure is .]niform, \\)l~cn trcatinb ;IIO siukprobe us a point sink it is ncccssary to Mitw an cflrctive proberadius, The cf~cctivo probe rwlius is ii point u distance ?,c uwayfrom the point sink at whi{h a prcssur(, corresponding trJ solutionof the corrccl proldcm and tho itpproximritc prohlwn tsrc ~qual,Sllorlna ulld 1)usstsn;17 ctilrulot ml where this point would IN if Ihowellbore wor~~[0 Lo ccmsidorod us huvihg an illfinito rudius, Ac.

];i the Il,llf spare ;~l)l~roxill)ul.itltlording !O tl~o work of \VilkiusollOf Sllnrlll ii Ulid l) UWUII37,.Is only vtilid for rclntivcly ;isw VUIUCSof tlw uniso[ro]~,v (say < 4 ) and lhut ut higlwr nuimtm]~y tlwcurwituro of the well Lccoulcs iu)lwrtantt

A Icading order StCISi~j-StiitC solu tion to the mixvd lJoul]d-isry Viduo proldom rcsultiag from llii\~illg ii holv (o\cr which thprossurv is uniforln) in al! inlpcrmrmlll~~ cylinder was givot~ by\Vilkil)son:13 i,s

.

(Itl)

rul ,0, :

(1:2)I)y uquuting Eusl 12 ulltl El tlm cftec(ivc proi.w radius lr cul~bc dcterimlumxl us ss f uuctiuu d ut)isot mpy and wellborv rudiu#,l$ig, 9 BINMVSU Idol Of lc Us. kr/k# .

Prom tlw zvro urdor solutlon of \Vilklllswi :) 111(!L4Yvltl\1~wellbore radius tur u poiill rslnk ill n half qmcc is

(u:])

1his i~ the solution dctcrmincd !)YSharma and l)us.ssm,uswl with IJq, IN it provides a good approximation uuriisotropy of 100. lhi~ is bccausc Eqo IY2 implicitly inclwellbore, As anisotropitw in SJXMWSf this would most l ikefrom t ho prcwncv of ~oyoring, in which case the malhomodel prwwnted here would I.winvalid, the zrwo order sofjcnerully adequate, If it is mxcssary to go to higher aoisthen tlm clycctive probe rtidills for tlw point cm wcylindedotfirmimd using the l)ol~llollliill

(s1 0.5563031577 0, J88W67!285fl~ 2h44042230fM 2,26103612s6(SS .2,3669107027 .2.1132!)602016] .(),()f)~l()$~~~~ .0,0750552906b~ .1,2417272971 . 1,060EW2073!I63 .1 ,WN)3!)331O .1,523WIW5

(1] 0.55s$1606s46 0,523M4S5Mu~ 2,4530G2SW2 2,S713212456(IJ .2,W)5(W51J) .2.i72t)i2i?)MIJl .o,(l!m3132Jti .oJ.)7?)mu!)73f)i .1,3103s13s0 .lJ64S37fi:)1563 .l,oti7LiWi&17 .1,!!0700088s3

-r:!1/)?U!q(1p4)/),

,-100 111(17fl (11)10 (III10 CII13/82X l@ l/l)si

c ])/),2(),336 Clll

222

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Pressnre D iffe r en c e, p s i

.=.=.=.==.=-=.==.=.=.==+-== =.=.=.=.=~=.= :.===------- -------- ------- x x7. -,-, =.:. :. L---- . . . . ...=. ~.~.l.~.~~.: .~.~.~.--- -! . , - ,. - ;- -- - ,._;-2- .7 = -,. ,: + L: -~+t. ,+~~~ ~.. _= ;---?,=::-==--- - - -J - .~,-- - aL. ==__ :7 .==F.,,=.,. r- _------ Y.-, ==&. Y.,.-, -

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1001l\.,8.0 . -.~0! 0 0.2 0.4 0.6 0.8 1,0

Spherical Time, sc#5Fw,ta4 1,1 1, b ,4 l :, f h ,1,,. L,\b: . 101.3.,., . ,. .s r ,~, .1, , , 1, . . 2 . , ,>.1. !

Thi ( h-s: 200 cm .

YTllicktw5s..?OOcm

Thickncssx w

o 20 40 60 80 100Time, sec

1,%1,,,.I(la,*,I% ,, ,4 41 1 ,s ,, ,, >, 1 ,, ,, . a ,, , I I ,. , ,, ! ,< ., 1 , ,1 ,. . , ,, , t ,. ,, , ,, . ! .< .

20T-.~~I 102 100 lL

I , ..), ! !, s,! , 1) .!.,,, .1, ,,. (,. ,,.,. .,. , f,,,,, ,, ,,, ,,( ,1,$. . , ,,, , $,,,,, ,, .1,.,,,,!!. !,!

2,0

1.6

1,2~~u

0.s

0.4

0.0

0.0

O=n

,

,00PEflona ..r Illickncs~ u 100rm25.0 -

.=B,g 20,0&&~Q 15,0 -S!~- 10,0 ,,, ,., ., . . . . . . . . . . . .

5,0

0,0 ~~Jo 20 40 60 80

Time, scc

I .Mt , 1 . I !1. . 9. .1 ,. ,81 d I c.u! ,, la ,, .. . .t , , 1 ,, .1 ,. ,, ,, , , ,, ,! , > !,

0, 8

0.6+%h0.4

0.2

0. 0 ~~10 10 102