SPESPE 22729

Practical Solutions for Interactive Horizontal Well Test Analysisc.u. Ohaeri and D.T. Vo, Unocal Science & Technology Div.SPE Members

Copyright t 991, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dallas, TX, October 6-9, 1991.

This paper was selected for presentation by an SPE Program Committee following 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 of 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

This study presents new approaches to solve problems offlow to horizontal wells. It documents a procedure toobtain solutions to problems of single phase liquid flow tohorizontal wells. This procedure, which couples the sourcefunction approach with a real space Laplace inversionalgorithm, gives very accurate results for a wide variety ofwellbore and reservoir conditions including constantwellbore pressure, wellbore storage, skin and fluid phaseredistribution. This approach is valid during both transientand boundary-dominated flow periods. Furthermore, thesolution technique is very fast and is ideal for interactivewell test analysis and design. This paper presents acomplete analysis of well test data using an interactivetype curve matching procedure and the solutions derivedin this study.

INTRODUCTION

Over the past few years, several studies have presentedsolutions to the problem of flow to a horizontal well.1- 5,7,8

The majority of the solutions pertain to the response of awell produced at a constant sandface rate in either aninfinite or closed rectangular reservoir. Generally, thesesolutions employ one of three solution procedures toderive an expression for the pressure distribution in thereservoir. The most widely used technique involves thesource function method of Gringarten and Ramey.6 Thisapproach readily yields results for the constant sandfacerate, but does not easily give solutions for either wellborestorage or constant sandface pressure production.

References and illustrations at end of paper

665

The second solution procedure is found in reference 7.This approach combines the Fourier and Laplacetransformation techniques. The resulting solution iscomplex. It is also difficult to use directly since it requiresusing both Fourier and Laplace inverse transforms.

Finally, in Ref. 8, one finds another solution method.Here the solutions exist in the Laplace space; they arederived from a combination of the source functionapproach and the Laplace transformation technique. Moreimportant, unlike the solutions in Refs. 1-5,7, Ref. 8presents analytic expressions for constant sandfacepressure production. Also, because the solutions are in theLaplace space, one can readily generate results forwellbore storage and skin.

In the course of this work, we evaluated several horizontalwell solutions and solution techniques presented in theliterature. We seek to develop a solution algorithm foruse in interactive well test analysis. Therefore, ourevaluation criteria are speed and accuracy. Because ourapplication is microcomputer based, for us, the idealsolution must be fast and accurate. It should also becapable of accounting for different inner and outerboundary conditions such as wellbore storage, skin, fluidphase redistribution and closed reservoir boundaries. Inanalyzing well test data over the years, our experience isthat these phenomena are nearly always present.

Although the solutions we examined produce accurateresults, they are not necessarily optimal for interactivewell test analysis. In all cases, these solutions arecomputationally slow. The source function derivedexpressions are particularly slow at early times. This is sobecause the equations involve sums of infinite serieswhich converge very slowly at small times. Since this

2 PRACTICAL SOLUTIONS FOR INTERACTIVE HORIZONTAL WELL TEST ANAL{SIS SPE 2272'9

approach requires integration from time zero, the slowconvergence of the series is a computational handicap.Similarly, the Laplace space based solutions, Refs. 7 and8, also involve infinite series summations which convergeslowly in some time domain. In addition, the lattersolutions require numerical inversion which increases thecomputional time since each time step involves severalevaluations of complex functional expressions.

Given this computational time problem, the present workaims to develop time-efficient horizontal well pressureand flow rate solutions which account for wellbore effects.This paper presents our solution. Although the pressuresolutions arise from a mixing of the source functionsolutions with those derived from method of images, oursolution algorithm and code optimization techniquesensure efficient computation. Furthermore, by applyingthe convolution integral to the pressure solution in theLaplace space, we derive results for wellbore storage,fluid phase redistribution and constant wellbore pressureproduction. We employ numerical schemes for bothLaplace transformation and inversion. Our Laplacetransformation algorithm is due to Roumboutsos andStewart.9 For Laplace inversion, we use the algorithm ofStehfest.10

PRESSURE SOLUTIONS

The dimensionless pressure at any point in a closedrectangular reservoir caused by single phase flow to asingle horizontal well is

{t +2E [emf- n2~2'tHmtJwD)coJ n1tJD)jL't (1)

/I-I ~l JeD JeD "1 JeD rTo demonstrate the accuracy and speed of our solutions,we have presented comparisons of our results with thoseobtained using the methods of Ref. 8. In terms ofaccuracy, the two solutions are always identical. In termsof speed, our solution technique, in the worst case,computes at least one order of magnitude in time fasterthan those of Ref. 8. In other cases, our technique isseveral orders of magnitude faster.

THEORETICAL CONSIDERATIONS

The following definitions apply:

Dimensionless pressure drop,

••••••••(2)

Dimensionless time,

Dimensionless distance in the x-direction,

Dimensionless well length,

........(5)

.......(3)

........(4)

PHYSICAL MODEL

Figure 1 shows a schematic of a horizontal well. Thereservoir is rectangular and has lateral dimensions x., , y.and height h. The well length is L and its center is locatedat distances, xy" Yw' and :lw from the origin. This permitsthe well location to be varied in three dimensional spacewithin the confines of the reservoir. In addition, thereservoir has permeabilities kx, !y, and kz in the x, y, and zdirections respectively. The reservoir porosity, cp, isconstant and we assume the flow of a single phase slightlycompressible fluid

Mathematically, the horizontal well is modeled as a linesource well. However, to account for the radius of actualhorizontal wells, we compute the sandface pressure at adistance IW away from the line source. Furthermore, thewellbore flow model assumes a uniform flux well. To getthe infinite conductivity well model results, we computethe well pressure as suggested by Gringarten and Ramey.6

666

SPE 22729 c. U. OHAERI AND D. T. VO 3

Dimensionless distance in the y-direction,

Y =2yD L

Dimensionless distance in the z-direction,

........(6)

swiftly and the series converges within a few terms.

These different convergence characteristics lead to anefficient algorithm for computing horizontal well pressureresponse. By using Eq. 9 at small times and Eq. 1 at largetimes, one can construct a solution algorithm thatconverges reasonably fast in all time ranges. Thisswitching algorithm is part of our overall computationprocedure.

To account for anisotropy, the horizontal permeability, k,is

Eq. 1 is derived using source functions.6 An alternativeexpression derived from the method of images is

Hl+';;wDl+e-{l-";;wDlH...(10)

p#"".....t,) =~lLtD{1+2!r exp{-n21t2L~'C~os(n1tZWD)cos(mtZD)}

The algorithm described above is fast but not optimal: Toachieve even greater computational speeds, we resort tothe solutions for the infinite reservoir. Our strategy is touse this solution during the transient flow period when itis identical to the bounded reservoir solution. Thedimensionless pressure at any point in the infinitereservoir created by single phase flow to a singlehorizontal well is given by

........(8)

.......(7)zz =-D h

By using the method of images, the alternative solution is

nE~ [err -(JD+YwD+2nyerl)+er( -(JD-YwD+2nYerl)] d'Cn.-~ lI.. 4'C Pt, 4'C 'C

..(9) H1+";;wD1+ ef-:j;wDlH..(11)

Although Eqs. 1 and 9 give identical results, they havedifferent convergence characteristics. Eq. 1 convergesrapidly for large values of the time group, 104t Eq. 9converges very slowly in this time range. Conversely, Eq. 9converges very rapidly for very small values of 1OI..o2 whileEq. 1 converges very slowly in this time range. Theseresults can be deduced if one considers the exponentialterms in both equations. In Eq. 1, large values of theabove time group cause the exponential term to approachzero rapidly leading to a convergence of the infinite serieswithin a few terms. Similarly, for small values of the timegroup, the exponential terms in Eq. 9 approach zero

To our knowledge, Eq. 11 has not heretofore beenpresented in the literature.

Eq. 10 is the analog of Eq. 1 while Eq. 11 is analogous toEq. 9. More important, Eqs. 10 and 11 converge faster inthe same time range than their counterparts, Eqs. 1 and 9.By switching from Eqs. 9 and 11 for small 101..02 to Eqs.11 and 1 for large 10I..o2

, one can achieve tremendouscomputational efficiency. This approach allows us tocompute pressures at any point in the reservoir in a timeefficient manner.

667

4 PRACTICAL SOLUTIONS FOR INTERACTIVE HORIZONTAL WELL TEST ANALYSIS SPE 22729

That is not all. If wellbore pressures are required, then byemploying the short time approximation to Eq. 10, we canfurther accelerate computation. This expression is givenb?

p"(Ix.I<I,y",,,,). 4-:'.' -~:] (12)

where:

and

I -e-II

-Ei(-x)= -du" u

........(13)

computer. Fig. 2 also contains the reservoir and wellparameters used. It plots the time required to computethe well dimensionless pressure response within a timeinterval, (tol-t,to,) as a function of the time group tolo2.Even though lo = 1 in Fig 2, the results for Eqs. 1, 9, 10and 11 correlate as a function of tolo2

• This graph istherefore a quantitative measure of computation speed.

Fig. 2 reveals the convergence characteristics of thevarious solutions including those of Ref. 8. As expected,Eq. 1 converges slowly when tolo2 is small but rapidly forlarge values. On the other hand, Eq. 9 has the exactopposite behavior. Also, the behavior of Eqs. 10 and 11parallels those of Eqs. 1 and 9. We also note theextremely small computation times for Eq. 12. Withregards to Ref. 8, we find that its solution converges fasterthan Eq. 1 at early times but is slower than the otherequations. More important, it is slower than all the othersolutions at late times.

Eq. 14 defines the dimensionless wellbore radius.Furthermore, Eq. 12 defines the early time vertical radialflow periodt-4 and is identical to Eq. 10 prior to tom'ngiven by

COMPUTATION ALGORITHM FOR PRESSURESOLUTION

The results of Fig. 2 indicate that one can construct anefficient algorithm to compute horizontal well pressures.The strategy here is to exploit the convergencecharacteristics of the various solutions in order to derivean algorithm which is equally fast for the infinite andclosed reservoirs. The procedure is as follows:

1) Use Eq. 12 to get the early time pressureresponse until:

i) One log cycle to the end of the early radialflow period, i.e to :S 0.1 tom'n' or

3) When cases 1 and 2 are no longer applicable, useEqs. 1 and 9 and switch between them based onthe magnitude of tolo2

• As described earlier, useEq. 9 for large values of tolo2 and Eq. 11 for smallvalues of the same time group.

In evaluating Eqs. 1, 9, 10, and 11, numerical integrationis required. The optimal computational scheme is todivide the time domain into a sufficient number ofintervals {too, tot, .... ton}' evaluate the most efficientexpression within each time interval (tol-t, to,), and sumthe integral for the final result. This procedure alsoaccelerates computation because if the previous resultsare stored in a pressure table, then at time tol we have

(l-xr)2

20(ZD-Zw~2

tDmin ~ min 20L2 (15)D

(ZD+ZWD-2)2

20L2D

Subject to the restrictions imposed by Eq. 15, Eq. 12provides a fast solution for sandface pressures at earlytimes. This is so because the equation involves only oneexponential integral. Furthermore, we note that thisexpression is valid for both the uniform flux well (XWo =0) and the infinite conductivity well ("we = 0.732). Wenote also that if lo is small and/or if rweis small, tomln willbe small. However, since practical values of lo and I"woare1O-t :s lo :S 500 and 1O-6:s 1"wo:S l(J"2, Eq. 12 is useful inpractically all cases. The alternative which involves Eqs. 1,9, 10 and 11, also requires integrating from even smallertimes.

Fig. 2 graphically depicts the computation speeds of thevarious pressure solutions already presented. It also showsthe performance of the solutions of Ref. 8. We performedthese comparisons on an Everex 386/20 personal

668

2)

ii) to ~ tomln i.e Eq. 12 is no longer valid.

If to ~ tomln and transient flow still exists, use theinfinite reservoir solutions, Eqs. 10 and 11. If thetime group tolo2 is small, use Eq. 10. If this timegroup is large, use Eq. 11. Above all else, usethese equations only until one log cycle before thenearest boundary is felt.

SPE 22729 c. U. OHAERI AND D. T. VO

Numerical Laplace Transform

5

........(16)

where f(to) can be Eq. 1, 9, 10 or 11. Also, Ib (tol-1) is theresult at tol-1' It represents the (i-1)st entry in the pressuretable and is available at tol' Thus the only integrationperformed is from tol-1 to to."

We recommend a numerical integration scheme with atleast fourth order accuracy. Furthermore, the schemeshould perform a minimum of function evaluations. In thiswork, we use the ten-point Gauss-Legendre integration14

which evaluates the function exactly ten times.

WELLBORE EFFECTS

The solutions presented so far do not account forwellbore storage, skin and phase redistribution. To addthese effects, we resort to the superposition theorem orDuhamel's principle. The well pressure response, P.vo (to)created by a time dependent flow rate 'b (to) is given bythe convolution integral

where /Ib = time derivative of the dimensionless

pressure drop caused by constant sandfacerate production.

'b = dimensionless flow rate

Pwo = dimensionless pressure due to the variablerate sequence 'b (to).

Eq. 17 can be solved either by directly evaluating theintegral or by using the Laplace transform approach. Wechoose the latter approach because the resultingexpressions are easier to manipulate. Thus, by applyingthe Laplace transform to Eq. 17, we get

PwD(S)= sqD(S)PD(S) •..•....(18)

Direct Laplace transformation of the pressure solutionsEqs. 1, 9, 10 and 11 is difficult. We have elected to use anumerical Laplace transformation procedure.9 In thisapproach, we first compute two arrays of dimensionlesstimes and pressures spanning the time of interest. Thistable is then transformed into the Laplace space using thealgorithm presented in Ref. 9. Thus, for the time andpressure tables {to1' lin, ···ton} and {lb1' 1b2'···· Ibn} wecompute the numerical Laplace transform from

.(20)

where

.......(21)

Wellbore Storage, Skin and Phase Redistribution

In this section, we develop solutions for wellbore storage,skin effect and fluid phase redistribution. This willenhance the utility of our horizontal well model inpractical well test analysis. The mathematical modelwhich describes constant wellbore storage, and fluid phaseredistribution at the well is13

~dPWD dP~D)

qD(t~ = 1 - C -tit--tit (22)D D

whereP;o = the dimensionless phase redistribution

parameterCo = the dimensionless storage constant

and

........(23)

The Laplace transform of Eq. 22 is

Here, s is the Laplace space variable and P.vo(s) is theLaplace transform of Ib (to) given by

........(24)

........(19)

669

In addition, if skin exists at the wellbore, thedimensionless pressure drop with skin but excludingwellbore storage and phase redistribution, Ib* (to), isgiven by

6 PRACTICAL SOLUTIONS FOR INTERACTIVE HORIZONTAL WELL TEST ANALYSIS SPE 22729

........(25)

where S is the skin factor.

1)

2)

Compute a table of dimensionless pressures andtimes as described in the pressure computationalgorithm section.

Numerically transform the table of dimensionlesspressures and times into the Laplace space usingEq.20.

Taking the Laplace transform of Eq. 25, we get 3) Use Eq. 27 or 28 for the desired solution in theLaplace space.

If we use Eqs. 24 and 26 in Eq. 18, and simplify theresulting expression we get the dimensionless wellborepressure with storage, skin and phase redistribution. Thisexpression is

END EFFECTS

In using the numerical Laplace inversion algorithm, Eq.21, the results at either end of the time range tend to bein considerable error. However, this error disappearsapproximately one log cycle from either end. To avoidthis problem, we compute the pressure results one logcycle before the minimum time and one log cycle past thelargest time. We recommend this procedure.

Invert the Laplace space solution in step 3 intothe real space using a numerical Larclaceinversion algorithm such as Stehfest 0 orCrump.11

4)

........(27)

........(26)-. - SPD(S)= PD(S)+ ­

S

CONSTANT SANDFACE PRESSURE PRODUCTION VALIDATION OF RESULTS

We now consider the case of constant sandface pressureproduction. Here, we are interested in the flow rateversus time response principally to generate declinecurves. From practical considerations, these curves areuseful in horizontal performance prediction and ratedecline analysis.

If we rearrange Eq. 18, and if we note that Pwo(s) = 1 / sin this case, the dimensionless well flow rate for constantsandface production becomes

........(28)

SOLUTION ALGORITHM FOR CONSTANT WELLPRESSURE, WELLBORE STORAGE AND PHASEREDISTRIBUTION

This section compares our model results with thoseobtained using the methods of Ref. 8. This will help tovalidate our model and demonstrate the speed of ouralgorithm. We present comparisons for constant wellpressure and for constant well rate with and withoutwellbore storage. Note that we wrote the program togenerate the results of Ref. 8. We also validated thisprogram against results presented in the paper.

In all comparisons, we use the following reservoirparameters: XeD = 100, YeO = 100, Xwo = 50, Ywo = 50, 7wo= 0.5 and IWo = 5 X 10-5 • This implies that the well iscentrally located in three dimensionless space. We alsopresent comparisons for two values of the dimensionlesswell length, I.o: I.o = 1 and I.o = 100. In all cases, thewell produces from the early transient to the boundarydominated flow period. Also, the solid lines represent ourmodel results while the open circles depict the results weobtained using the method of Ref. 8.

Given the foregoing derivations, we have developed ageneral procedure to "add" wellbore storage, skin andphase redistribution effects to the horizontal well constantsandface rate pressures using Eq. 27. To generate theconstant sandface pressure solution, we use Eq. 28. Moreimportant, this procedure is general in nature; it appliesto other reservoir systems besides those with horizontalwells. Here, we present the algorithm as it pertains to thehorizontal well case. The procedure is as follows:

Fig. 3 presents a comparison of dimensionless pressures.It compares both the dimensionless pressure and itsderivative with respect to the natural logarithm of time.The agreement between our model results and those ofRef. 8 is excellent during all flow regimes. This is alsoshown in Table 1. This table presents a numericalcomparison of the dimensionless pressures 'in Fig. 3. Inaddition, Fig. 4 presents the same comparison for twovalues of the dimensionless wellbore storage constant, Co,Co = 0.1 and Co = 1. Finally in Fig. 5, we compare theresults for constant sandface pressure production. Asdescribed previously, our model results in Figs. 4 and 5

670

SPE 22729 c. U. OHAERI AND D. T. VO 7

come from the application of the convolution integral tothe constant sandface rate solution by way of thenumerical Laplace transformation technique. In thesefigures, there is excellent agreement between our resultsand those of Ref. 8.

We now compare computation speed. Table 2 shows thespeed comparison for the results in Figs. 3, 4, and 5obtained using an Everex 386/20 computer. It presentsthe time required to compute 11 log cycles using 15 pointsper log cycle. The results in Table 2 clearly indicate thatour model is much faster in all cases. For thedimensionless pressures, our model uses 2.44 seconds. Forwellbore storage solutions, it uses 15.8 seconds. Finally,for the constant wellbore pressure case, our model uses15.6 seconds. The comparable time for Ref. 8 in all thesecomparisons is about 540 seconds.

The preceding results demonstrate the speed and accuracyof our horizontal well model. It is obvious that this modelcan be used to generate accurate horizontal well solutionsfor interactive well test analysis using a personalcomputer. This is very useful because, in addition to toand other reservoir parameters, horizontal well responsesdepend on I"wo and ~ at early times. Therefore, it is notpossible to generate a priori a comprehensive set of typecurves which will cover all situations. Nevertheless, onceI"wo is known, one can generate a set of type curves where~ becomes the varying parameter and this type curve canthen be used to analyze the pressure and rate data ofinterest. Alternatively, one can use this model directly toestimate well and formation properties using a parameterestimation algorithm.

TYPE CURVES

Fig. 6 presents horizontal well type curves generated usingour model. We present this figure as an example. Thefigure shows the parameters used The dimensionless welllength, ~ is the parameter varied. The solid linesrepresent the dimensionless pressure while the dashedlines depict derivative of the same pressures with respectto the natural logarithm of time. We note that initially theresponses depend on~. This result is reportedelsewhere.2 After the start of semi-steady flow, to > 1<1,the various curves merge into one.

Figure 7 presents horizontal well decline curves obtainedusing our model. These curves pertain to the constantsandface pressure case. The reservoir parameters areidentical to those of Fig. 6 and ~ is the parameter ofinterest. Here again, the early time rate behavior dependson ~. Also, after the onset of reservoir depletion, to >1<1, the curves for ~ ~ 10 merge together. However, thecurves for ~ < 10 do not follow the same decline curveas do the curves for ~ ~ 10. This result was unexpectedand, to the best of our knowledge, it is not reportedelsewhere in the literature.

671

FIELD EXAMPLE

This section presents the analysis of field pressure buildupdata. The objective of this test was to estimate formationproperties and also to evaluate the well condition. Thehorizontal well length is 388 ft. Table 3 contains otherrelevant data used in the analysis. In addition the wellproduced for 4320 hours prior to shutin.

The analysis presented here is based on type curvesgenerated using our model. Fig. 8 shows some of thedimensionless pressure type curves used as well as theirderivatives. These are drawn using solid lines. Thedimensionless well length, ~, is 0.868. Furthermore, thetype curves account for wellbore storage and fluid phaseredistribution. The dimensionless wellbore storageconstant, Co, is 1, the apparent dimensionless storageconstant, Cao, is 5 and the dimensionless phaseredistribution pressure, P;o, is 0.2. The selection of theseparameters required numerous iterations in which eachparameter was varied while keeping everything elseconstant. This process is carried out interactively on apersonal computer. Even though one can use anestimation algorithm for this task, nevertheless, thegraphical approach gives us better control over theparameters of interest. The speed of our solutionalgorithm makes this possible. In Fig. 8, the parametervaried is the skin factor, S.

The well pressure response shows the influence ofwellbore storage, skin and phase redistribution. From thematch, we get the following results: k = 3.7 md and S = ­2. Using ~ = .868, the permeability ratio, kz/k = 0.05.This gives a vertical permeability of 0.185 md. Althoughnot shown here, results obtained from semi-log analysis ofthe radial flow period in Fig. 8, at> 10 hours, agree withthose presented here.

The above results demonstrate that, even when wellborestorage and skin effects are present, one can estimatehorizontal well parameters from well test data using typecurves. However, this method requires having data beyondthe wellbore storage region so that the dimensionless welllength, ~, can be determined independently. We notethat if only one radial flow period is evident, then semi­log analysis alone will give ambiguous results since it willbe difficult to determine which radial flow period isrepresented. On the other hand, the type curve matchingapproach takes into account all prevailing flow regimes.

CONCLUSIONS

This work has examined different solutions for singlephase flow to a horizontal well. Based on the precedingdiscussions and results, the following conclusions are inorder:

8 PRACTICAL SOLUTIONS FOR INTERACTIVE HORIZONTAL WELL TEST ANALYSIS SPE 22729

1)

2)

3)

This study has developed a fast and efficientalgorithm to compute horizontal well pressures foruse in interactive well test analysis using apersonal computer.

This study has also presented solutions forhorizontal well flow rates which are useful fordecline curve analysis.

In this study, we have demonstrated the use of anumerical Laplace transform algorithm togenerate constant sandface pressure solutions aswell as constant sandface rate solutionsincorporating wellbore effects in cases wheresuch solutions would not otherwise be available.This approach is general. It can be applied toother reservoir systems.

p; = phase redistribution pressure, psia

P;o = dimensionless phase redistribution pressure

q = flow rate, STB/ day

<b = dimensionless wellbore flow rate

IW = wellbore radius

ro = dimensionless radial distance

rwo = dimensionless wellbore radius

S = skin factor

s = Laplace space variable

NOMENCLATURE

B = Formation volume factor, Res vol/vol

Cy, = phase redistribution parameter, psi

h = formation thickness, ft

t = time, hours

to = dimensionless time

teq = equivalent flow time, hour

z = distance in the z-direction, ft

Xc = dimensionless distance in the x-direction

y = distance in the y-direction, ft

Yo = dimensionless distance in the y-direetion

x = distance in the x-direction, ft

La = dimensionless distance in the z-direction

J1. = fluid viscosity, cp

¢ = formation porosity

We have demonstrated the use of our horizontalwell solution algorithm in interpreting fieldpressure buildup data by way of interactive typecurve matching on a personal computer.

4)

c = wellbore storage constant, Res BBI / psi

<aD = apparent dimensionless storage coefficient

Co = dimensionless wellbore storage constant

Cy,o = dimensionless phase redistribution parameter

G = total compressibility, l/psi

k = horizontal permeability, md ACKNOWLEDGEMENTS

REFERENCES

We are indebted to the management of Unocalcorporation for permission to present this study.

kx = permeability in the x-direction, md

ky = permeability in the y-direetion, md

kz = permeability in the z-direction, md

L = horizontal well length, ft

In = dimensionless horizontal well length

p = pressure, psia

fu = dimensionless pressure

A = initial reservoir pressure, psia

1)

2)

Clonts, M. D. and Ramey, H. J., Jr.: "PressureTransient Analysis For Well With HorizontalDrain Holes," paper SPE 15116 presented at the1986 California Regional Meeting, Oakland, April2-4.

Ozkan, E., Raghavan, R., and Joshi, S.D.:"Horizontal Well Pressure Analysis," SPEFE (Dec.1989) 567-75; Trans., AIME, 287.

Pwo = dimensionless wellbore pressure

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SPE 22729 c. U. OHAERI AND D. T. VO 9

3) Adalberto, J. R, and Carvalho, R:. "AMathematical Model for Pressure Evaluation ofInfinite-Conductivity Horizontal Well," SPEFE(Dec. 1989) 559-666.

4) Odeh, A S., and Babu, D. K: 'Transient FlowBehavior of Horizontal Wells: Pressure Drawdownand Buildup Analysis," SPEFE (March 1990) 7-15.

5) Daviau, F., Mouronval, G., Bourdarot, G., andCurutchet, F.: "Pressure Analysis for HorizontalWells," SPEFE (Dec. 1988) 761-724.

6) Gringarten, A C., and Ramey, H. J., Jr.: 'The Useof Source and Green's Functions to SolveUnsteady-Flow Problems in Reservoirs," Soc. Pet.Eng. J. (Oct. 1973) 285-296; Trans AIME, 255.

7) Goode, P. A and Thambynayagam, R K M.:"Pressure Drawdown and Buildup For HorizontalWells in Anisotropic Media, " SPEFE (Dec. 1987)683-697,; Trans. AIME, 283.

8) Ozkan, E. and Raghavan, R: "Some NewSolutions To Solve Problems in Well TestAnalysis: Part I - Analytical Considerations," paperSPE 18615, presented at the SPE Joint RockyMountain Regional/Low Permeability ReservoirsSymposium and Exhibition, Denver Colorado,March 6-8, 1989.

9) Roumboutsos, A and Stewart, G.: "A Direct De­convolution or Convolution Algorithm for WellTest Analysis," paper SPE 18517 presented at the63rd Annual Technical Conference and Exhibitionof the SPE, Houston, Texas, Oct, 2-5, 1988.

10) Stehfest, H.: "Numerical Inversion of LaplaceTransforms," Communications of ACM, 13, No.1(Jan.1970) 47-49.

11) Crump, K S.: "Numerical Inversion of LaplaceTransforms Using a Fourier SeriesApproximation," Journal of the ACM, (Jan. 1976)89-96.

12) van Everdingen, A F. and Hurst, W.: "TheApplication of the Laplace Transformation to FlowProblems in Reservoirs", Trans. AIME, 186, 305­324.

13) Fair, W. B., Jr. : "Pressure Buildup Analysis WithWellbore Phase Redistribution," Soc. Pet. Eng. J.(April 1981) 259-270.

14) Press, W. H., Flannery, B. P., Teukolsky, S. A, andVetterling, W. T.: Numerical Recipes, NewYork, Cambridge University Press, 1986, 121-125.

673

TABLE 1'SPE 2 272 ''I

comparison of Dimensionless Pressure Solutions

[xeo Yeo 100, x wo Ywo = 50, Zwo = 0.5, r wo 5 x 10-5 ]

1.0= 1 1.0 = 100

t D Dimensionless Pressure, PD

This Study Ref. 8 This Study Ref. 8

1.E-06 1. 7002 0.01701.E-05 2.2758 0.02291.E-04 2.8514 2.8514 0.0350 0.03501.E-03 3.4270 3.4270 0.0734 0.07341.E-02 4.0027 4.0027 0.1946 0.19461.E-01 4.5889 4.5889 0.5761 0.57611.E+OO 5.4772 5.4771 1.4620 1. 46191.E+01 6.5925 6.5925 2.5773 2.57721.E+02 7.7401 7.7401 3.7249 3.72481.E+03 8.9431 8.9406 4.9279 4.92551.E+04 14.610 14.645 10.595 10.5941.E+05 71.159 67.1441.E+06 636.65 632.63

TABLE 2

Computational Time 9 Log-Cycles, Secs

[xeo Yeo 100, X wo = YWf) = 50, Zwo = 0.5, It = 1, r wo = 5 x 10-5 ]

This Study Ref. 8

Fig. 3, PD' CD = 0 2.41 539.92

Fig. 4, PD, CD =0.1 15.77 539.2

Fig. 4, PD, CD =1.0 15.76 539.1

Fig. 5, Clo 15.56 553.25

TABLE 3

Data Used in Analysis

Initial reservoir pressure, Pi

Reservoir temperature, T

Formation thickness, h

Porosity, rp

oil formation vol. factor, Bo

oil viscosity, ~o

Total compressibility, ct

oil flow rate, go

Producing time, tp

Horizontal well length, L

wellbore radius, r w

674

1500 psia

210 Deg. F

50 ft

0.15

1.0 Res. BBI/STB

1.0 cp

10-6 psia

200 STB/D

4320 ft

388 ft

0.25 ft

10610510410.410.4 10.3 10.2 10.1 1 101 102 103

Dimensionless Time Group, tDL2D

j10~~IxwD=100 YwD=50

i . ~YwD = 100 rwD = 5 x 10-4xwD = 50 zwD = 0.5

(J)

LD=1

ap "Ea;;- = 0

i=

" Eq.10a.

.1! 10.1 ~I-

".2§ 10.2 "/:>a.E Eq.9

8 10.3 Eq.15

"

HorizontalWell

Figure 1 - Cut Away Section of Horizontal Well Producing a Closed RectangularReservoir

Figure 2 . Comparisons of Computational Time for Various Solution Techniques

~

Figure 3 • Horizontal Well Pressure and Pressure Derivati~e Responses,Comparison to Ozkan and Raghavan Solutions

en-0IT1

N610

N~

I~

~

YwD = 50 XeD = 100zwD = 0.5 YeD = 100rwD =5x105 xwD =50

~'LOGARITHMICSLOPEdpwo'dlntD

CONSTANT SAND FACE RATECLOSED EXTERNAL BOUNDARIESCLOSED BOnOM BOUNDARY000 OZKAN AND RAGHAVAN-- THIS STUDY

10·3~, """If, "",,,1 ,,,,,,,,1 ,,,,, .. ,1 ",,,,,,1 ,,,,,,,I ,,,.,,,, """,,' "',,' """,'

10.5 10-4 10.3 10.2 10.1 10 0 10 1 10 2 103 10 4 10 5

DIMENSIONLESS TIME, tD

101

Figure 4 • Horizontal Well Pressure and Pressure Derivative Responses,Wellbore Storage Effect. Comparison to Ozkan and Raghavan Solutions8

102

" ",,,,,, """" """'" """'" "'"'''' """'" """'" """'" """'" "''''., "''''.

c~;:.5

c.'O.0Ul ;:ll: C.::>'0(J)l\! ~ 10

0

a.O(J)..J(J)(J)

~2z2!~ E 10.1

zll:Ul«2!gc..J

ll:o

10.2XeD = 100

YeD = 100xwD = 50YwD = 50

zwD = 0.5rwD =5X10·5

CD =0

CONSTANT SAND FACE RATECLOSED EXTERNAL BOUNDARIESCLOSED BonOM BOUNDARY000 OZKAN AND RAGHAVAN-- THIS STUDY

L O = 1

10 ·2

103

10·3i ,.,.",,1 """"i """,I "",,,,1 """"I "''''',' ",,,,,,I "",,,, '''''',,' """,I ,,,,,,I

10.5 10-4 10.3 10.2 10.1 100 10 1 10 2 10 3 10 4 10 5 10 6

DIMENSIONLESS TIME, tD

104

t: j, 1111111 I,Llliil i I I ""'1 Iii ,lilt! 1 i i "ILII i [,lIitll [, lillil! [i i 111111 Iii ""'I ,'i II ,ill I 1111181

c61:l~ 10 2W ;:lea.~'O(J)Ul

l\! ~ 101

~iil ~(J)(.lUl-..J::;;

i5 ~ 100

we:z«UlCl::;;0is ~ 10.1

o

104

E """"1 """'I ""'''I ''''''''1 ''''''''1 ''''''''1 ''''''''1 """"1 ''''''''1 ''''''''1 ''''''~ 104~ """"1 "'''''I ''''''''1 ''''''''1 ''''''''1 ''''''''I ''''''''1 ''''''''1 ''''''''1 """"1 """1I II

XeD = 100YeD = 100xwD =50YwD = 50

zwD = 0.5rwD = 5E·4CD =0

dP DCONSTANT WELLBORE RATE -- LOGARITHMIC SLOPE*CLOSED EXTERNAL BOUNDARIES n DCLOSED BOTTOM BOUNDARY ------- PwD

La = 0.1

100r----~-----~_> ..---:,- •.... -- ..... -..... -

101

102

103

10 -31 I t "lI"l , ",,!Ii ' II ",I !" 'II' I ,t1 tllll! !! "II I " "",I !" "11,1 I •• ,1 "!!,,I! 1 111l,,1

10.5 10-4 10.3 10.2 10.1 10 0 10 1 10 2 10 3 10 4 10 5 10 6

DIMENSIONLESS TIME, tD

10.1~ .... __ ._---~:: ..~- -y

10 ·2

C·EC­;:""

"-Cui ;:0:"­::>""Ul ww"­0: 0,,-oJUlUlUlO

~§!!z:I:01::iii!riffiCl:159Co:

o

xeD = 100 YwD = 50YeD = 100 zwD = 0.5xwD =50 rwD =5xl0·5

La =1

CONSTANT WELLBORE PRESSURECLOSED EXTERNAL BOUNDARIESCLOSED BOTTOM BOUNDARY000 OZKAN AND RAGHAVAN-- THIS STUDY

101

100

102

103

10.31 ,,,.,,,! '''!!''! .""",1 II ""! '!!!I11'! '""II'! ''''II'! "".1 'IlU"! ,!~

10.5 10.4 10.3 10.2 10.1 100 10 1 10 2 10 3 10 4 105 10 6

DIMENSIONLESS TIME, tD

10·1

10.2

ol:T

ui~0:UlUlWoJZoiiizw:0is

~

Figure 5 - Horizontal Well Rate Decline Responses, Comparison to Ozkan andRaghavan SolutlonsB Figure 6 - Horizontal Well Pressure and Pressure Derivative Responses

'Cf)"'tJfT\

~

~J

'"N~

~!lQ.

l~oa

102

Equivalent Time, 8teql hours

Figure 8 • Field Test Example: Horizontal Well Pressure Buildup DataInfluenced by Wellbore Storage and Phase Redistribution

"[g­~

\ 102

i(;

/

XeD = 100 YwD = 50YeD = 100 zwD = 0.5xwD = 50 rwD =5 x 10.4

CONSTANT WELLBORE PRESSURECLOSED EXTERNAL BOUNDARIESCLOSED BOTTOM BOUNDARY

10 ·31 ", "",I ", ",,,I "" ",I '" ",,,I """,,1 '" ",,,I '" ",,01 '" "",I '" "",I " \\1",1\ I" ,,,,,I10.5 10-4 10.3 10.2 10.1 100 10 1 10 2 103 10 4 105 10 6

DIMENSIONLESS TIME, tD

10 ·2

410

103

102

0l:T

~10

1c(0:UlUlWoJ

100z

0iiizw:0is

Figure 7 - Horizontal Well Rate Decline Responses