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Abstract This paper presents experimental and theoretical investigations of multilateral/duallateral horizontal well performance in areally anisotropic reservoirs. The investigation is based on a 3D analytical solution. The analytical solution considers an arbitrary orientation of the multilaterals/duallaterals in horizontal plane.

Experimental study was conducted on an electrical analog apparatus. Experimental data was compared with the results from the analytical productivity model. Excellent agreement was obtained between the results.

The multilateral well model was also compared against a model presented in the literature and verified.

Introduction Duallateral completion is the most common practice of multilateral drainhole technology. It is generally believed that the productivity of horizontal wells is proportional to the well length. However, as the well length increases, drilling and well control become more difficult. Additionally, transportation of large volumes of fluid along a long horizontal borehole results in considerable pressure losses in the wellbore. Wellbore pressure loss yields a decrease in well productivity. Multilateral wells provide an alternative to drilling long single horizontal wells. The advantages of multilaterals are;

• More contact area with reservoir • Accelerated production • Better well control compared to a long single lateral • Less drill sites • Less drilled footage • Minimization of wellbore pressure losses • Better reservoir coverage and definition • Improved control over flooding patterns

Many publications on the modeling of the flow into multilaterals have appeared in the literature. The previous work could be grouped in three categories; productivity models1-6, transient-flow models6-14, and field applications.15-19 Prediction of production performance of multilateral wells has been discussed by Brand et al.1, Salas et al.2, Economides and co-wrokers3-5, and Madden and Vo.6

The works on transient-flow models investigated the transient pressure and derivative responses of and rate distribution along multilateral wells. Karakas et al.7 used a numerical simulator to analyze the well test data from a duallateral well in a carbonated reservoir in Middle East. Using a semianalytical procedure, Raghavan and Ambashta8 computed the pressures and derivatives of various multilateral well configurations. A heuristic discussion of the flow regimes and the corresponding slopes of the pressure vs. time plots are also provided in the same study. Later, Ozkan et al.9 investigated the characteristics of pressure, derivative, and rate distribution in duallateral horizontal wells. Some later studies10-14 concentrated on the transient rate responses as well as transient pressure behavior of multilateral wells.

Case studies on the evaluation of multilateral well performance have also been presented by Madden and Vo6, Karakas et al.7, Hall15, Longbottom16, Lee et al.17, Stalder et al.18, and Yaliz.19

As discussed in the literature, duallateral/multilateral wells exhibit complicated flow characteristics. In some cases, responses of duallateral horizontal wells are analyzed by using an equivalent single horizontal well. This approach can be valid for certain phase angles and anisotropy ratios but may lead to significant errors in estimation of well productivity for many duallateral/multilateral well configurations.

The objectives of the current study are 1) to develop a 3-D model to predict the productivity of multilaterals, 2) to verify the model against the experimental data, and 3) to investigate the productivity of multilaterals under steady state flow conditions.

Mathematical Model Fluid flow into multilateral wells displays a complicated geometry and requires a 3D physical model to simulate the flow into each branch. The model considered in this study is sketched in Fig. 1. A multilateral well in a finite rectangular parallelepiped reservoir is considered. A steady state flow condition is assumed. The reservoir is sealed at the top and bottom of the formation. All four lateral surfaces exhibit constant external pressure. The reservoir is homogeneous but anisotropic. xk , yk , and zk are the principal permeabilities

SPE 94223

Multilateral Horizontal Well Productivity T. Yildiz, SPE, Colorado School of Mines

2 SPE 94223

coinciding with the directions of the Cartesian coordinate system. A single-phase fluid of constant viscosity and compressibility is assumed.

The analytical solution considers an arbitrary orientation of the multilaterals in the horizontal plane. However, the geometry of the laterals in the vertical plane is restricted; all the laterals are drilled parallel to the sealed boundaries at the top and bottom of the reservoir. The laterals are represented as line sources. The laterals deviate (in x-y plane) from the direction of the principal permeability xk (the permeability in the x-direction) by an angle of wiθ . Each branch may have a different length, diameter, and phasing angle. A variable skin factor, owing to formation damage, along each lateral is also considered.

The mathematical details of the multilateral well model producing under steady state flow are given in Appendix A. The solution is obtained by using the finite Fourier sine and cosine transforms. The final mathematical model is in the form of a matrix. The solution of the matrix yields the pressure drop and fractional rate distribution of the laterals.

A computer code was developed to solve the matrix given in Eq. A-34. The model and the code are verified against the results from the analytical model presented by Salas et al.2 and the experimental data obtained on an electrolytic apparatus. The experimental study and model verification are described below.

Experimental Study There exists an analogy between the single-phase fluid flow in porous media and the current flow in an electrical field. Therefore, the single-phase fluid flow problems may experimentally be simulated on electrolytic models by using the analogy. In the past, many problems involving fluid flow in porous media have been investigated using the electrolytic models.20-24 Experimental studies on electrolytic models are cost effective, fast, and easier to control. The earlier studies on the modeling of flow into perforations, inclined wells, hydraulic fractures, and complicated well geometries relied on experimental results from electrolytic analog apparatus. See Refs. 20 through 24. Apparatus. In this study, we extended the use of electrolytic models to investigate the productivity of duallateral and multilateral horizontal wells. A schematic of the experimental setup used in the investigation is given in Fig. 2. More details on the setup and experimental procedure can be found in Ref. 24. The apparatus is basically composed of a 100x100x50-cm Plexiglas tank, a variable voltage source, and several ammeter/voltmeter pairs. The Plexiglas tank was filled with brine of known salinity and conductivity. Copper plate electrodes were mounted inside the Plexiglas tank to introduce the electricity to the apparatus. Several well models made of copper wires and plates were manufactured. The wells were placed at different locations in the tank as well as at the center of the tank.

To examine the effect of polarization and hysterisis, experiments were repeated with different salinities and potential drops. Independent resistivity measurements on the brine samples were also conducted using a commercial

resistivity-meter. To investigate the effect of polarization and hysterisis, a fully penetrating vertical well was placed at the center of Plexiglas tank and current was measured at several potential drop values. The apparent resistivity of the medium was calculated using the radial form of Ohm’s law. The results indicated some minor hysterisis and polarization effects. In a set of experiments, we typically measured the current at 10, 20, 30, and 40 Volts of potential drops.

Experimental Results. The experimental results obtained from the electrolytic apparatus are presented for duallateral and chicken-foot configuration. For the experimental data presented below, the coordinate system is located at the bottom-left-front corner of the rectangular parallelepiped reservoir model.

Duallaterals. Many experiments on duallaterals have been run on the experimental setup. Four sets of experimental data on duallaterals are documented in Table 1. The geometry of the duallateral well configuration is displayed in Fig. 3. The first and second laterals are 29-cm and 20.2-cm long, respectively. Both laterals have a radius of one mm. The locations of laterals in each experiment are listed in Table 1. Notice that the configuration in experiment #4 can be also considered as a multilateral with four drainholes. The measured resistances are tabulated in Table 1.

Multilaterals. Compared to the modeling of multilaterals emanating from the same vertical section, the analytical modeling of multilaterals with chicken-foot configuration is more complicated and requires special treatment. To verify the analytical solution for the multilaterals with chicken-foot geometry, additional experiments have been conducted. In the chicken-foot configuration, four branches emanate from the main horizontal drainhole. The geometry of the chicken-foot configurations is given in Fig. 4 and Table 2. The main drainhole length is 30.2 cm. The lengths of branches #2, #3, #4, and #5 are 7, 7, 7.1 and 7.2 cm, respectively. The angles between the main horizontal drainhole and branches #2, #3, #4, and #5 are +44, -47, +24, and –24 degrees, respectively. The measured values of resistance for the chicken-foot geometry are listed in Table 2. Model Verification The analytical multilateral well model presented in this study has been compared against the experimental data described above and the results from an analytical solution available in the literature.

Comparison with Experimental Data. The comparison is given in terms of resistance since the calculation of resistance from the analytical solution does not require the use of measured potential drop or current. The resistivity value measured on the commercial resistivity-meter was used in the analytical simulation of the experimental data.

The last column in Table 1 shows the resistance predicted by the analytical model for the duallateral geometry. A comparison of measured and predicted resistance values in Table 1 demonstrates excellent agreement.

The analytical multilateral well model has also been used to simulate the experiments conducted on the chicken-foot configuration. The predicted resistances for the chicken-foot

SPE 94223 3

geometry are tabulated in the last column of Table 2. For all three experiments, the simulated results agree very well with the measured values. The average difference between the simulated and measured resistance values is about 5%.

Comparison with Salas et al Model. Salas et al.2 presented results from a steady state multilateral well model. The results from their analytical model are presented in Figs. 1 and 2 of Ref. 2. They presented the results in terms of productivity ratio which is the productivity index of a 2,000-ft long multilateral/horizontal well divided by the productivity index of a 1,000-ft long single horizontal well. All the cases considered by Salas et al.2 are given in Table 3. For the sake of completeness, all the other data used by Salas et al.2 are also reported in Table 3.

The multilateral well model developed in this study was used to simulate the results for the well configurations described in Table 3. The results from this study are listed in the last column of Table 3. As can be seen on Table 3, an excellent agreement between the results from this study and the Salas et al.2 model are observed.

Acknowledgments Part of the study reported in this paper was completed when the author was with Istanbul Technical University. The study was funded by The Scientific and Technical Research Council of Turkey (TUBITAK) under YDABCAG-154.

References 1. Brand, C., Economides, M. J., Frick, T. P., Retnanto, A.:

“Optimal Configuration of Multiple-Lateral Horizontal Wells,” paper SPE 35712 presented at the 1996 SPE Western Regional Meeting, Anchorage, Alaska, May 22-24.

2. Salas, J. R., Clifford, P. J., and Jenkins, D. P.:“Brief: Multilateral Well Performance Prediction,” JPT (1996).

3. Economides, M.J., Brand, C.W., and Frick, T.P.:”Well Configurations in Anisotropic Reservoirs,” SPEFE (December 1996) 257.

4. Economides, M. J., Frick, T. P., and Smith, J.:“Reducing Economic Risk in Areally Anisotropic Formations with Multiple-Lateral Horizontal Wells,” paper SPE 30647 presented at the 1995 SPE Annual Technical Conference & Exhibition, Dallas, Texas, October 22-25.

5. Economides, M. J. and Retnanto, A.: “Performance of Multiple Horizontal Well Laterals in Low-to Medium-Permebility,” SPERE (1996).

6. Madden, M. V. and Vo, D. T.:“Performance Evaluation of Trilateral Wells: Field Examples,” SPERE (Feb. 1995).

7. Karakas, M., Yokoyama, Y. M., and Arima, E. M.: “Well Test Analysis of a Well with Multiple Horizontal Drainholes,” paper SPE 21424 presented at the 1991 Middle East Oil Show, Bahrain, November 16-19.

8. Raghavan, R. and Ambashta, A.K.: “An Assessment of Productivity of Multilateral Completions,” Journal of Canadian Petroleum Technology (October 1998) 58.

9. Ozkan, E., Yildiz, T., and Kuchuk, F.J.:”Transient Pressure Behavior of Dual-Lateral Wells,” SPEJ (June 1998) 181.

10. Basquet, R. et al: “A Semianalytical Approach for Productivity Evaluation of Complex Wells in Multilayered Reservoirs,” SPEREE (December 1999) 506.

11. Larsen, L.: “Pressure-Transient Behavior of Multibranched Wells in Layered Reservoirs,” SPEREE (February 2000) 68.

12. Ding, Y.: “Transient Pressure Solution in the Presence of Pressure Drop in the Wellbore,” paper SPE 56614 presented at the 1999 SPE Annual Technical Conference and Exhibition, Houston, Texas, 3–6 October.

13. Ouyang, L.-B. and Aziz, K.: “A General Single-Phase Wellbore/Reservoir Coupling Model for Multilateral Wells,” SPERE (August 2001) 327.

14. Yildiz, T.:”Multilateral Pressure-Transient Response,” SPEJ (March 2003) 5.

15. Hall, S.D.:“Multi-Lateral Horizontal Wells Optimizing a 5-Spot Waterflood,” paper SPE 35210 presented at the 1996 Permian Basin Oil and Gas Recovery Conference, Midland, TX.

16. Longbottom, J. R.:“Development and Testing of a Multi-Lateral System,” paper SPE 35545 presented at the 1996 European Production Operations Conference and Exhibition, Stavanger, Norway.

17. Lee, S.S., Veeken, C.A.M., and Frens, A.M.:”Multi-Lateral Well Modeling to Optimise Well Design and Cost,” paper SPE 68728 presented at the 2001 SPE Asia Pacific Oil and Gas Conference and Exhibition, Jakarta, Indonesia, April 17-19.

18. Stalder, J.L. et al.: “Multilateral-Horizontal Wells Increase Rate and Lower Cost Per Barrel in the Zuata Field, Faja, Venezuela” paper SPE 69700 presented at the 2001 SPE International Thermal Operations and Heavy Oil Symposium, Porlamar, Margarita Island, Venezuela, March 12-14.

19. Yaliz, A.: “Case Study of a Quad-Lateral Horizontal Well in the Lennox Field: A Triassic Oil-Rim Reservoir,” paper SPE 75249 presented at the 2002 SPE/DOE Improved Oil Recovery Symposium, Tulsa, Oklahoma, 13–17 April.

20. McDowell, J.M. and Muskat, M.:”The Effect on Well Productivity of Formation Penetration Beyond Perforated Casing,” Trans. AIME (1950) 189, 309.

21. McGuire, W.J. and Sikora, V.J.: “The Effect of Vertical Fractures on Well Productivity,” Trans., AIME (1960) 219, 401.

22. Roemershauser, A.E. and Hawkins, M.F. Jr.: “The Effect of Slant Hole, Drainhole and Lateral Hole Drilling on Well Productivity,” JPT (Feb. 1955) 11.

23. Tinsley, J.M. et al.: “Vertical Fracture Height-Its Effect on Steady-State Production Increase,” JPT (May 1969) 633; Trans. AIME, 246.

24. Yildiz, T. and Deniz, O.: “Experimental Study on the Productivity of Complex Well Configurations,” paper SPE 50433 presented at the 1998 SPE International Conference on Horizontal Well Technology, Calgary, Alberta, November 1-4.

25. Abramowitz, M. and Stegun, I.A.: Handbook of Mathematical Functions, Dover Pub., Inc., New York, 1972.

26. Gradshteyn, I.S. and Ryzhik, I.M.: Table of Integrals, Series and Products, Academic Press. Inc., London, UK, 1980.

Nomenclature Bo = formation volume factor, rbbl/stb h = height, ft k = permeability, md L= reference length Lh= lateral length nL= number of laterals p = pressure, psi q = flow rate, stb/day r = radius, ft rw = wellbore radius, ft rwiD = dimensionless equivalent lateral radius, ft sd = skin factor due to damage SD= source function xwi = heel of lateral i in x-direction , ft

4 SPE 94223

ywi = heel of lateral i in y-direction , ft zwi = vertical location of lateral i , ft θp = angle between the lateral axis and principal

permeability direction. µ= viscosity, cp Subscripts d = wellbore damage D= Dimensionless h= horizontal L= Lateral w= well x= x direction y= y direction z= z direction

SI Metric Conversion Factors bbl x 1.589 873 E−01 = m3 cp x 1.0* E−03 = Pa*s ft x 3.048* E−01 = m ft3 x 2.831 685 E−02 = m3 in. x 2.54* E+00 = cm lbf x 4.448 222 E+00= N lbm x 4.535 924 E−01 = kg mD x 9.869 233 E−04 = µm2 psi x 6.894 757 E+00 = kPa *Conversion factor is exact.

Appendix A - Mathematical Model A rigorous 3-D model considering the complicated geometry of multilateral horizontal wells has been developed. Although the solutions for transient flow in finite reservoirs with several different sets of boundary conditions have been derived, only the model for steady state flow, which can be used to estimate the productivity of multilateral wells, will be presented. A sketch of the multilateral well model is depicted in Fig. 1. The solution is developed in terms of dimensionless variables. The 3-D diffusivity equation governing the steady state flow into a multilateral with Ln laterals producing at a constant flow rate is:

0),,(2

2

2

2

2

2=+

∂∂

+∂∂

+∂

∂DDDD

D

D

D

D

D

D zyxSzp

yp

xp (A-1)

In Eq. A-1, SD is the source function for a multilateral well and can be expressed as follows:

iwiDDhiD

n

1iDDD I)zz(

L1kh2S

L

−= ∑=

δπ (A-2)

where

u)~sinuyy()~cosuxx(qI wiwiDDwiwiDD

L

0iDi

hiD

dθδθδ −−−−= ∫ (A-3)

The solution of Eq. A-1 is subject to the following

boundary conditions.

0)z,y,x(px,0x DDDDeDD == (A-4)

0)z,y,x(py,0y DDDDeDD == (A-5)

0)z,y,x(zp

h,0z DDDD

DDD =

∂∂

= (A-6)

It is also required that the sum (the integral) of the flux

along the laterals be equal to total well flow rate; that is,

1uqL

1 hiDL L

0iD

hiD

n

1i=∫∑

=

d (A-7)

The dimensionless variables are defined by

( )[ ]z,y,xppBq2.141

hkp eot

D −=µ

(A-8)

Lkkxx xD //= (A-9)

L/k/kyy yD = (A-10)

L/k/kzz zD = (A-11)

L/k/khh zD = (A-12)

yx2

D kk/kk = (A-13)

tsiiD q/L)t,z,y,x(qq = (A-14)

In the preceding definitions, L is a reference length in the system and k is the geometric mean permeability,

3 zyx kkkk = . In this study, L is chosen as below.

∑=

=Ln

1ihiL

21L (A-15)

Note that the above definitions transform the original

anisotropic system into an equivalent isotropic system but also alter the original lengths of the laterals and their angle from the positive x-direction. The dimensionless lateral lengths and locations are given by

L/k/kxx xwiwiD = (A-16)

L/k/kyy ywiwiD = (A-17)

SPE 94223 5

wi2

ywi

2

x

hihiD sin

kkcos

kk

LL

L θθ += (A-18)

( )wiyxwi tank/karctan~

θθ = (A-19)

A dimensionless lateral radius is also defined as follows.

( ) ( )[ ]25.0zwi

25.0wiz

wiwiD k/kk/k

L2r

r += (A-20)

where

wi2

xwi2

ywi sinkcoskk θθ += (A-21)

The solution of the boundary value problem defined above can be obtained by applying finite Fourier sine transformation to the x and y coordinates and finite Fourier cosine transform to the z coordinate. The general solution is given as below.

mlimli1m ml

ml

1l

n

1i hiD

iD

eDeD

DDDDDD I

RLq

yxhk4)z,y,x(p

L

Ψξ

π ∑∑∑∞

=

∞

==

= (A-22)

)y/ylsin()x/xmsin(R eDDeDDml ππ= (A-23)

)h(sh)]zz(h[ch)]zz(h[ch

Dml

wiDDDmlwiDDDmlmli ξ

ξξΨ

+−+−−=

(A-24)

du)]~sinuy(ylsin[)]~cosux(

xmsin[I wiwiD

eDwiwiD

eD

L

0mli

hiD

θπθπ++= ∫

(A-25)

2eD

2eD

2ml )y/l()x/m( ππξ += (A-26)

Refs. 25 and 26 document the explicit expressions to

evaluate the integral in Eq. A-25. The solution in Eq. A-22 assumes uniform-flux distribution along each lateral. This assumption yields variable pressure along the laterals. An infinite-conductivity condition should be imposed to have a uniform pressure along each drainhole. Integrating the uniform-flux solution along the jth lateral approximates the infinite-conductivity condition (the pressure averaging method). Evaluating Eqs. A-22 through A-25 for the jth lateral at wjDwjDjD

~cosrxx θ+= , wjDwjDjD

~sinryy θ+= , and

wjDwjD*jD rzz += , and using pressure-averaging method, we

obtain,

D*jDjDjD

L

0jD

hjDwjD r)z,y,x(p

L1p

hjD

d∫= (A-27)

mljmlimlij1m ml1l

n

1i hjDhiD

iD

eDeD

DDwjD II1

LLq

yxhk4p

L

Ωξ

π ∑∑∑∞

=

∞

==

= (A-28)

)h(sh)]zz(h[ch)]zz(h[ch

Dml

wiD*jDDmlwiD

*jDDml

mlij ξξξ

Ω+−+−−

=

(A-29)

The skin effect due to formation damage around each lateral could be incorporated into the solution by using the infinitesimally thin skin concept. It is considered that the flow around the laterals in the damaged zone is normal to the drainhole axis. Therefore, the additional pressure drop due to formation damage around each lateral is expressed as

djwjzhj

ojsj s

kkL

Bq2.141p

µ∆ = (A-30)

wj

dj

damwjz

wjzdj r

rln)1

)kk(

kk(

Lhs −= (A-31)

where sdj is the skin factor including the effect of permeability reduction in damaged zone around the jth lateral. damwjz )kk( and rdj are the permeabilities and radius of damaged zone, respectively.

The dimensionless solution including skin effect is:

hjDdjjDwjDwsjD L/sqpp += (A-32)

1qLn

1iiD =∑

=

(A-33)

Writing Eq. A-32 for each lateral, requiring the pressure at

each lateral be equal to the wellbore pressure, wsDp , and coupling the pressure equations with the rate equation (Eq. A-33), nL+1 equations with nL+1 unknowns are obtained. These equations can be written in a matrix format as below:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

00001

q.

qqp

a.aa1.....

a.aa1a.aa1

1.110

Dn

D2

D1

wsD

nn2n1n

2n2221

1n1211

LLLLL

L

L

(A-34)

where

jiji Aa = (A-35)

hjDdjjjjj L/sAa += (A-36)

mljmlimlij1m ml1lhjDhiDeDeD

DDji II1

LL1

yxhk4A Ω

ξπ ∑∑

∞

=

∞

=

= (A-37)

6 SPE 94223

TABLE 1- EXPERIMENTAL DATA FOR DUALLATERALS h = 20 cm, rw = 0.1 cm, re = 56.4 cm, ρ = 22.8 Ω m, Lh1 = 29 cm, Lh2 = 20.2 cm

Measured Parameters Predicted

Experiment #

Location of Lateral 1

Location of Lateral 2

I (current) (Ampere)

∆V (Volt)

R (Ohm)

R (Ohm)

1 xw1=28 yw1=52.5 zw1=11.8 θw1= 0 o

xw2=33 yw2=17.5 zw2=5.5 θw2= 5o

0.184 0.378

0.577 0.756

9.89 20.02 30.2 39.5

53.8 53 52.3 52.3

49.5

2 xw1=28.2 yw1=49 zw1=12 θw1= 0 o

xw2=28.5 yw2=44 zw2=7 θw2= 28o

0.151 0.321 0.499 0.646

9.61 20.11 30.8 39.9

61.8 61.7 62.6 63.6

58.9

3 xw1=28.5 yw1=49 zw1=11.8 θw1= 0 o

xw2=39 yw2=48.7 zw2=7 θw2= 297o

0.149 0.309 0.477 0.631

9.71 19.8 30.2 39.8

65.2 64.1 63.3 63.1

60.6

4 xw1=31.5 yw1=50.3 zw1=12 θw1= 0 o

xw2=40.2 yw2=44 zw2=12.8 θw2= 90o

0.140 0.290 0.454 0.593

9.62 19.69 30.5 39.7

68.7 67.9 67.2 66.9

65.3

TABLE 2- EXPERIMENTAL DATA FOR MULTILATERALS

h = 20 cm, rw = 0.1 cm, re = 56.4 cm ρ = 24.4 Ω m Lh1 = 30.2 cm, Lh2 = 7 cm, Lh3 = 7 cm, Lh4 = 7.1 cm, Lh5 = 7.2 cm

Measured Parameters Predicted Experiment

# Location of

main drainhole I (current) (Ampere)

∆V (Volt)

R (Ohm)

R (Ohm)

1 xw1= 20.2 yw1= 50.3 zw1= 0.1 θw1= 0 o

0.0981 0.192 0.308 0.424

9.98 19.12 30.08 40.7

101.7 99.8 97.7 96

98

2 xw1= 20 yw1= 20.6 zw1= 0.1 θw1= 0 o

0.108 0.221 0.340 0.449

10.02 20.08 30.4 39.9

92.8 90.9 89.4 88.9

91

3 xw1= 20.5 yw1= 20 zw1= 0.1 θw1= 44o

0.103 0.211 0.320 0.433

10.01 20.12 30.0 40.3

97.2 95.4 93.8 93.1

94.8

SPE 94223 7

TABLE 3 – COMPARISON OF MULTILATERAL WELL MODELS

rw = 0.354 ft, re = 5,000 ft, Bo = 1.35 RBBL/STB, µ= 0.4 cp, h = 100 ft, kx = ky =100 md, kz =10 md, Lh = 1,000 ft (Case 1), Lh = 2,000 ft (Cases 2-9)

Productivity ratio, fraction

Case Well configuration Salas et al. This study

1

1.00 1.00

2

1.46 1.45

3

1.36 1.36

4

1.33 1.33

5

1.26 1.27

6

1.22 1.22

7

1.21 1.22

8

1.2 1.21

9

1.17 1.18

8 SPE 94223

Fig. 1- Multilateral well model.

AC Source

Voltm eter

Am m eter

100 cm

50 cm100 cm

Fig. 2 – Electrolytic analog apparatus.

y

x

(xw1 ,yw1)

Lh1=29 cm

(xw2 ,yw2)

θw2

Lh2= 20.2 cm

Fig. 3 – Flow configuration for duallaterals.

y

x

9.9 cm 10 cm 10.3 cm

7 cm

7 cm

7.1 cm

7.2 cm

12

3

4

5

Fig. 4 – Multilateral (chicken-foot) well geometry.

wiy

y

x

yk

xk

dis

hiL

wix

wiθ

wiy

y

x

yk

xk

yk

xk

dis

hiL

wix

wiθ

ex

ey

h

x

yz