Production-Systems Analysis for Fractured Wells James L. Hunt, SPE, Halliburton Services

Summary. Production-systems analysis has been in use for many years to design completion configurations on the basis of an expected reservoir capacity. The most common equations used for the reservoir calculations are for steady-state radial flow.

Most hydraulically fractured wells require the use of an unsteady-state production simulator to predict the higher flow rates associated with the stimulated well. These high flow rates may present problems with excessive pressure drops through production tubing designed for radial-flow production. Therefore, the unsteady-state nature of fractured-well production precludes the use of steady-state radial-flow inflow performance relationships (IPR's) to calculate reservoir performance. An accurate prediction of fractured-well production must be made to design the most economically efficient production configuration.

It has been suggested in the literature that a normalized reference curve can be used to generate the IPR' s necessary for production-systems analysis. However, this work shows that the reference curve for fractured-well response becomes time-dependent when reservoir boundaries are considered. A general approach for constructing IPR curves is presented, and the use of an unsteady-state fractured-well-production simulator coupled with the production-systems-analysis approach is described. A field case demonstrates the application of this method to fractured wells.

Introduction Production-systems analysis has been used for many years to de-sign completion configurations on the basis of an expected reser-voir capacity. Often called nodal-systems analysis, this approach has been applied to the analysis of electrical circuits and pipeline systems. GilbertI was one of the first to propose the application of the systems-analysis approach to well producing systems.

A typical producing system includes many components where there is a potential for pressure drops to occur. As the well config-uration becomes more complex, the potential for large total pres-sure drops within the system increases. Fig. 1 presents a schematic of a producing configuration and possible pressure drops through the system.

It is the objective of production-systems analysis to optimize the well configuration for maximum production capacity. This is ac-complished by dividing the system at some point or node and cal-culating pressure drops within each component. System components upstream of the node are commonly referred to as the inflow; those downstream of the node are referred to as the outflow. Relation-ships between pressure drop and flow rate must exist for each com-ponent. Pressure drops for various flow rates are calculated for both inflow and outflow sections. Two conditions are necessary for pro-duction systems analysis: (1) flow into the node must equal flow out of the node and (2) only a single pressure can exist at the node for a given flow rate.

With these two conditions satisfied, flow capacity of the entire system can be determined. This is commonly achieved by plotting node pressure vs. flow rate for both inflow and outflow; the inter-section of inflow and outflow curves is the system flow capacity. This is illustrated in Fig. 2. The effect of a change in any of the components can be investigated by recalculating either the inflow or outflow curve, depending on the location of the component to be changed. For example, if the component is located in the out-flow section, the outflow curve is recalculated; however, the in-flow curve remains unchanged. Thus, the production-systems-analysis approach can be used to evaluate existing producing sys-tems and to aid in the design of future well configurations. Many examples illustrating the application of the production-systems-analysis technique exist in the literature. 2-6

To apply the systems-analysis approach to a certain well config-uration, relationships or models must be available for determining the pressure drop as a function of flow rate for each component considered. For calculation purposes, the well configuration can

Copyright t 988 Society of Petroleum Engineers

608

be separated into several sections or modules, each containing sever-al components. For example, the producing system shown in Fig. 1 can be separated into three general sections: flowline, well, and reservoir. Each of these sections may be composed of one or more components. For example, the well module may consist of a tub-ing string composed of several different sizes, a restriction within the tubing near the bottom of the hole, and possibly a safety valve that also introduces a flow restriction. Models that relate pressure drop to flow rate within a component are used to calculate the total pressure drop at a given flow rate for each section.

Correlations for multiphase flow through pipelines are available in the literature and are useful in determining pressure drops through tubing and flowline components of the well system. Correlations commonly used for calculating pressure drop through a horizontal pipeline include Refs. 7 through 10. Various pressure-drop corre-lations are available for flow through the vertical tubing of a well. 7,11-14 Pressure-drop relationships for flowline and tubing sec-tions have been in use for many years and are generally the accept-ed models for flow through pipes. Selection of one correlation over another depends on specific well conditions.

The reservoir is one of the most important components of the total system because it determines what will flow into the bottom of the well and is the most complex component of those studied in a well system. Consequently, the reservoir must be accurately described by an appropriate model. Various reservoir models, com-monly called IPR's, have been described in the literature; most deal with steady-state radial flow. For oil wells, these include Vogel's 15 equation, Standing'sl6 modification of the Vogel equation, Fetkovich'sl7 equation, and the familiar radial form of Darcy's equation. For gas wells, the common IPR's are the backpressure equation and Darcy's radial-flow equation. These IPR's are ade-quate in most cases for determining pressure drops through the reser-voir. For hydraulically fractured wells, however, especially long fractures and tight formations, the steady-state radial-flow IPR's are not adequate because of the unsteady-state nature of fractured-well flow.

Several methods of dealing with stimulated wells have been sug-gested. One method involves a modification of the existing steady-state radial-flow IPR equation by changing the flow efficiency to represent the stimulated condition. 16 This approach is limited in that flow efficiency, a steady-state concept, does not account for the unsteady-state response of fractured wells. A second method involves the use of published production-increase curves, such as those described by McGuire and Sikora 18 and Soliman. 19 This sec-

SPE Production Engineering, November 1988

C==~=---~ SALES GAS LINE

SEPARATOR

LIQUID

~P1 = PR - Pwf. LOSS IN POROUS MEDIUM

Pwf-pwh

BOTTOM HOLE RESTRICTION t

LOSS ACROSS COMPLETION ~P2 = Pwf. - Pwf

~P3 = PUR - POR

~P4 = PUSV - POSY

RESTRICTION

SAFETY VALVE

~P3= _POR' ~P5 = pwh - posc SURFACE CHOKE

(PUR-POR) It T-PUR

~P6 = Posc - Psep IN FLOWLINE

~P7 = Pwf - Pwh TOTAL LOSS IN TUBING

FLOWLINE

Fig. l-Producing-configuration schematic and possible pressure drops through the system.

Flow Rate

Fig. 2-Systems-analysis plot showing reservoir-inflow and tubing-outflow curves.

ond approach is also limited because the production-increase curves do not take time-dependence into account. A more involved ap-proach, presented by Meng et al.,20 uses a fractured-well model to develop the IPR curve for the stimulated well. Several fractured-well models have been presented in the literature. 21-23

Description of FracturedWell Model The fractured-well model used in this work was presented by Soli-man et al. 23 The model consists of a well intercepting a finite-conductivity vertical fracture producing at a constant flowing pres-sure under unsteady-state conditions. The fracture extends an equal distance on either side of the wellbore and fully penetrates the for-mation in height. The well is located in a limited reservoir consist-ing of a square drainage area with no-flow outer boundaries. Additional assumptions include the following.

1. Homogeneous and isotropic formation is of constant height. 2. Gravity effects are negligible. 3. The reservoir fluid is single-phase and compressible.

SPE Production Engineering, November 1988

10'

10'

Dimensionless Time

Fig. 3-Fractured-well type curve for e tD = 0.2.

4. Fluid flow in both the fracture and the formation is described by Darcy's law.

Solutions to the problem just described are presented as type curves. The following definitions are used in the development of type curves:

qBp. qD= , ................................. (1)

kh(Pi-Pwj)

kt tLjD=--2' ................................... (2)

cf>p.ctLf

and

Cf CjD=-' ...................................... (3)

1rkLf

609

TABLE 1-WELL AND RESERVOIR PARAMETERS FOR FIGS. 4 THROUGH 6, 8, AND 9

o

Formation permeability, md Formation thickness, ft [m) Porosity Initial reservoir pressure, psi [MPa) Wellbore radius, in. [mm) Drainage radius, ft [m) Reservoir temperature, of [0C) Gas specific gravity

0.10 32 [9.8)

0.107 2,394 [16.51)

4 [102) 2,640 [805)

260 [127) 0.65

~:~~-.-.. ---... -... -... -.. --.. -.----~I.'=.'~' ~~;'~~~~~'~~;:~':'i~'

~g ", i- ...... .

i:'\\\ 50 100 150 200 250 300 350 400 4!iO 500

Flow Rote (mel/d)

FIg. 4-IPR curves generated wIth Darcy equation.

Type curves are presented as plots of dimensionless flow rate vs. dimensionless time with the ratio of the fracture half-length to reser-voir extent as a parameter. Each type curve represents a specific value of dimensionless fracture conductivity. An example type curve for a CjD value of 0.2 is presented in Fig. 3. These plots are use-ful in predicting flow rate decline over time for a fractured well. Use of type curves for predicting fractured-well performance has been demonstrated. 21 23

Because of the number of type curves and the number of param-eters that describe the type curves, it becomes much easier to use a computer program to calculate the fractured-well performance than to perform the necessary calculations by hand. To facilitate ease of use, type curves are stored in the computer as continuous functions. An appropriate interpolation scheme is used to interpo-late between curves. The resultant type-curve simulator is able to calculate fractured-well performance with less computation time than would be necessary with a numerical simulator.

Fractured-Well Inflow As mentioned previously, many common IPR's describe steady-state radial flow. An example of this type of IPR equation is the radial form of Darcy's law, which can be used to generate an IPR curve for an unstimulated well. A hydraulically fractured well can be accounted for by use of the radial Darcy's-law IPR equation and by calculating an equivalent skin based on fracture half-length with the following equation24 :

Lf =2rwe-s . ...................................... (4)

For the well described in Table I and a fracture half-length of 400 ft [122 m], the equivalent skin, s, is -6.4. The resultant value of skin factor may be substituted into the Darcy's-law IPR to yield the reservoir inflow performance curve for the stimulated case. Ap-plying the Darcy equation IPR to a gas well described by the pa-rameters listed in Table 1 for both unstimulated and stimulated cases yields the inflow performance curves presented in Fig. 4. Com-parison of the two curves shows that the inflow performance curve generated for the hydraulically fractured case yields higher flow rates at the same drawdown, as would be expected for a stimulated well.

610

~ - - - ~ - ~ - - -Flow Rote (mel/d) Fig. 5-IPR curves generated with steady-state productlon-Increase model.

250 500 750 1000 1250 1500 1750 2000

Flow Rot. (mel/d)

Fig. 6-Tlme-dependent IPR curves generated with fractured-well simulator.

Application of the steady-state production-increase curves 19 to the example gas well (Table 1) results in Fig. 5. The unstimulated IPR curve was generated with the Darcy equation, and the production-increase curves were used to determine the folds of in-crease in production for the case of a 400-ft [i22-m] fracture half-length and a CjD value of to. The IPR curve resulting from the production increase calculation is presented as the stimulated case in Fig. 5.

A comparison of Figs. 4 and 5 shows that the stimulated cases yield similar results. The IPR's used to generate Figs. 4 and 5, how-ever, are based on steady-state equations; the effect of flow-rate decline over time, as observed in the field and predicted by the-ory, is not taken into account. Thus, the common IPR's that describe fractured-well response in relatively simple terms do not adequately describe observed fractured-well performance over the life of the well.

Applying the described type-curve simulator to the fractured gas well presented in Table 1 produces the IPR presented in Fig. 6, based on a 400-ft [122-m] fracture half-length and a CjD value of to. Several curves are presented for different times in the life of the fractured well. As time increases, the calculated flow rate at a given drawdown decreases, as expected. This gives rise to the expected flow-rate decline over time for a well produced at con-stant bottomhole flowing pressure (BHFP). In addition, the fractured-well model predicts much higher flow rates than that pre-dicted with the radial form of Darcy's law with a negative skin factor and steady-state production-increase curves. This is expected be-cause of the unsteady-state nature of fractured-well flow.

SPE Production Engineering, November 1988

~ -;

Q.

co ci

'" ci

ci

N ci

Reservoir Type: Uquld Flow Two Pha now Ga. flow

O~-------r-------r-------.-------'------~ o 0.2 0.4 0.6 0.8

q/qMAX

Fig. 7-IPR reference curve proposed by Vogel. 15

co ci

'" ci

Production x months after frac: )( = 1 x = 30 x.60

O~-------r-------r-------.-------.------~ o 0.2 0.4 0.6 0.8

q/qMAX

Fig. 8-Gas-well IPR reference curve: without effect of reser-voir boundaries.

Generation of IPR Curve. For solution-gas-drive wells, Vogel 15 presents a correlation for ob-taining IPR curves. A plot of Pwflji R vs. qlqrnax was proposed as a reference curve to generate IPR's. Fig. 7 shows a plot of the refer-ence curve for different reservoir types. From this plot, it is evi-dent that a straight-line relationship exists only for single-phase liquid flow.

Extending the concept of the straight-line reference curve to gas wells, it has been proposed that a reference curve similar to that of Vogel could be obtained from which the IPR could be generat-ed. Meng et ai. 20 proposed that use of real-gas pseudopressures25

to plot PpwtfPjJ R vs. q(t)lqmax(t) obtains a straight-line relationship that holds throughout the entire producing life of the gas well. From

SPE Production Engineering, November 1988

co ci

Production x months after frac: x = 1 x = 30 x = 6Q

O~------'--------r-------.-------'------~ o 0.2 0.4 0.6 0.8

q/qMAX

Fig. 9-Gas-well IPR reference curve: Includes effect of reser-voir boundaries.

TABLE 2-WELL AND RESERVOIR PARAMETERS FOR FIGS. 10 AND 11

Formation permeability, md Formation thickness, ft [m) Porosity Initial reservoir pressure, psi [MPa) Wellbore radius, in. [mm) Drainage radius, ft [m) Reservoir temperature, OF [0C) Gas specific gravity Oil gravity, API [g/cm 3) GOR, scf/STB [std m3 /stock-tank m3 )

0.5 35 [10.7)

0.30 5,000 [34.47)

3.48 [88.4) 1,320 [402)

200 [93) 0.65

40 [0.825) 1,000 [180.1)

this proposed straight-line relationship, the time-dependent IPR curves can be generated.

The type-curve simulator described earlier was used to generate a normalized reference curve for the fractured gas well described by the properties listed in Table I without the effect of reservoir boundaries; all pressures were converted to pseudopressures. The resultant plot, presented in Fig. 8, shows that a relatively straight-line relationship results when pseudopressures are used and the ef-fect of boundaries is not considered; the reference curve also ex-hibits no significant time-dependence. When the effects of the reservoir boundaries are considered, however, the reference curve becomes time-dependent, as shown in Fig. 9. A significant devia-tion from the straight-line relationship exists, which increases with time. Thus, boundaries have a pronounced effect on the proposed reference curve. Consequently, when the effect of a closed drainage region on fractured-well performance is considered, the time-dependent IPR curves should be generated directly using the type-curve simulator to calculate flow-rate decline over time at various BHFP's. From the resultant output, flow rates at the various pres-sures are then plotted as a graph of BHFP vs. flow rate with time as a parameter. The resultant plot presents time-dependent IPR curves for the fractured well.

Effect of Stimulation and Tubing Size Production-systems analysis is useful in designing completion con-figurations for new wells. Table 2 presents well and reservoir pa-rameters for a proposed oil well. A systems-analysis plot for two different tubing sizes is shown in Fig. 10. The Darcy equation was

611

~fo~.~~[~u.~".~n.~,~=o~--------------------r.=~~~~~

o o o

o-!,----.---.---,---,----,---r---.---,-~._--~ o 10 20 30 ~o 50 60

Flow Rale (sIb/d) 70 BO 90 100

Fig. 10-Systems-analysls plot showing effect of tubing size: Darcy-equatlon IPR.

TI",.',"o TI",.',"o Tlm.18mo

~.~~el.~a~~!~~:,' . G 2.U2LD.Tublnl

100 200 300 .. 00 500 600 700 800 900 1000 1100

Flow Ral. (stb/d)

Fig. 11-Systems-analysls plot showing effect of tubing size: fractured-well-simulator IPR's.

used to generate the reservoir inflow and the Hagedorn and Brown 13 correlation was used to generate the tubing outflow curves. For the particular conditions presented in Table 2, the system-flow capaci7, for the smaller-diameter tubing is 72 STBID [11.4 stock-tank mId] but 63 STBID [10.0 stock tank m3 /d] for the larger-diameter tubing. To maximize production on the basis of information presented in Fig. 10, this well would most likely be completed with the smaller-diameter tubing. The type-curve simulator was used to generate the IPR curves presented in Fig. 11 to investigate the effect that a 4OO-ft [122-m] fracture half-length would have on the system flow capacity. From this plot, it is ap-parent that early in the life of the well, a larger-diameter tubing would be needed to obtain maximum production capacity. The difference in flow rates between the two tubing sizes at 1 month after fracturing is about 100 STBID [15.9 stock-tank m3/d], a sig-nificant difference. Later, however, it would be necessary to change to the smaller-size tubing to maintain maximum production capac-ity. In this case, through the use of a fractured-well simulator to calculate time-dependent IPR curves, the time at which the tubing should be changed to a smaller diameter could be estimated for the stimulated condition. This type of analysis is not possible when a steady-state inflow equation is used to calculate the reservoir per-formance.

Effect of Wellhead Pressure and Fracture Half-Length The production-systems-analysis approach has been applied to a south Texas gas well described by the well and reservoir parame-ters presented in Table 3. This well is typical of many of the gas wells drilled in that area. It is anticipated that at least an 800-ft

612

TABLE 3-WELL AND RESERVOIR PARAMETERS FOR FIGS. 12 THROUGH 15

Formation permeability, md Formation thickness, ft [m) Porosity Initial reservoir pressure, psi [MPa) Wellbore radius, in. [mm) Drainage radius, ft [m) Reservoir temperature, OF [0C) Gas specific gravity

o l,=800ft

0.030 12 [3.7)

0.18 3,950 [14.89)

3.94 (100) 2,640 (805)

180 (82) 0.73

~~------------------r=~~~ \, I \'

i \\, o I \,

~~ i \.\ a. ' \ "-- I .,

e i \ \. ~! \ \ fo"O! \\, Q." I \ " ~ i \ \ i ~----~-~---------- ---------- -\.._._._._.-.:.\_._\_._._._._._._._._._._._.- ._._. __ ._.-~ g ! \. ...

S! -r--------------\-------:------------------------------ ---I \ "

250

i \

500 750 1000

Flow Rate (mcf/d) 1250 1500

Fig. 12-Systems-analysls plot showing effect of wellhead pressure.

\ o \'

~ ! \\ I \ "

\ \ '. ~~ \ \\ ..e- I \ ... ~' \ " 3 g ! \ ... 5 N i \. \. .to i \ " -g~ I\.\' ~! \ \, ~ g ! \ \

I \ " i \ ...

~ i \ '. i i'"

o i \ ... ~+-----,-----,-----.-----,-----.-----.-----

200 400 600 800 1000 1200 ' .. DO

Flow Rat. (mel/d)

Fig. 13-Systems-flow capacity as a function of wellhead pressure and time.

[244-m] hydraulic fracture is needed to produce this well econom-ically. It is desired to determine the productive capacity of this well at various wellhead pressures for 1.995-in. [50.67-mm] -ID tubing.

Constructing IPR curves for the fractured case with the fractured-well type-curve simulator yields the time-dependent inflow curves presented in Fig. 12. The radial-inflow curve was calculated for the unstimulated case with the radial form of Darcy's law and is presented for comparison. Outflow curves were generated for var-ious wellhead pressures with the Cullender and Smith 14 correla-tion for gas flow. For simplicity, the flowline section was not considered.

Fig. 13 was constructed by plotting the intersections of the out-flow and inflow curves from Fig. 12 (production capacity at each wellhead pressure and time) as wellhead pressure vs. flow rate. The resultant plot presents system-flow capacity as a function of

SPE Production Engineering, November 1988

o It = 800 ft. P",,,, = 900 ptl

!l

0 0 0

0 ~o

~CQ .s

0

.!~ 0

Q:

~ 0 0 G:~

0

1l

0

. " 16 2. U 2. 32 3. Time (mo.) Fig. 14-Flow-rate profile for constant wellhead pressure.

wellhead pressure and time. From Fig. 13, the effect of producing at constant wellhead pressure can be determined by plotting flow rate vs. time at constant wellhead pressure. Fig. 14 plots predicted flow rate vs. time at constant wellhead pressure (900 psi [6200 kPaD for a fracture half-length of 800 ft [244 m]. This type of plot can be constructed for various constant wellhead pressures to determine the flow-rate-vs.-time profile for the fractured well.

This procedure can also be used to compare the effect of frac-ture half-length on producing at constant wellhead pressure. Fig. 15 presents the predicted flow-rate-vs.-time profiles at a constant wellhead pressure of 900 psi [6200 kPa] for fracture half-lengths of 800 and 1,200 ft [244 and 366 m]. The effect of several differ-ent fracture half-lengths can be investigated in this manner. Thus, by use of the production-systems-analysis approach coupled with a fractured-well simulator, production can be maximized for a given set of conditions for fractured wells producing under unsteady-state conditions.

Conclusions 1. The production-systems-analysis approach is useful in evalu-

ating existing producing systems and in the design of new well con-figurations.

2. IPR's generated with steady-state radial-flow models or pub-lished production-increase curves do not adequately model fractured-well performance. Because of the time-dependent nature of fractured-well response, production-systems analysis is accom-plished more effectively with a fractured-well simulator used to generate the IPR curves.

3. A fractured-well model that considers finite-conductivity ver-tical fractures and reservoir boundaries is useful in constructing time-dependent IPR curves for fractured wells.

4. The influence of reservoir boundaries on fractured-well per-formance causes the reference IPR curve proposed in the literature to be time-dependent.

5. The effect of a change in producing conditions on fractured-well response can readily be investigated through the application of production-systems analysis.

Nomenclature B = FVF, RBISTB [res m3/stock-tank m3] C t = total compressibility, psi - 1 [kPa -I ] Cf = fracture conductivity, md-ft [md' m]

CjD = dimensionless fracture conductivity h = formation thickness, ft [m] k = formation permeability, md

Lf = fracture half-length, ft [m] PDR = pressure downstream of flow restriction, psi

[kPa] PDSC = pressure downstream of surface choke, psi

[kPa]

SPE Production Engineering, November 1988

~ P""" = 900 psi

o o ~

o o

"

12 16 20 Time (mo.)

u 2.

Fig. 15-Flow-rate profile showing effect of fracture length at constant wellhead pressure.

P DSV = pressure downstream of safety valve, psi [kPa]

Pe = reservoir pressure at Xe, psi [kPa] Pi = initial reservoir pressure, psi [kPa]

PPR = real-gas pseudopressure, average reservoir, 106 psi2/cp [kPa2/Pa' s]

Ppwf = real-gas pseudopressure, wellbore flowing, 106 psi2 /cp [kPa2/Pa's]

ji R = average reservoir pressure, psi [kPa] Psep = separator pressure, psi [kPa] PUR = pressure upstream of flow restriction, psi

[kPa] PUSV = pressure upstream of safety valve, psi [kPa]

Pwf = BHFP, psi [kPa] Pwfs = BHFP at sandface, psi [kPa] Pwh = wellhead pressure, psi [kPa]

L!..pl' . llPs = component pressure drop, psi [kPa] q = flow rate, STB/D [stock-tank m3/d] or

MscflD [std m3/d] qD = dimensionless flow rate

qmax = flow rate at BHFP=O, STB/D [stock-tank m3 /d] or MscflD [std m3 /d]

rw = wellbore radius, ft [m] s = equivalent skin t = time, hours

tLjD = dimensionless time (fractured system) Xe = drainage distance, ft [m]

Jl. = fluid viscosity, cp [Pa' s] = porosity, fraction

Acknowledgment I thank the management of Halliburton Services for permission to prepare and publish this paper.

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Prod. Prac., API, Dallas (1954) 126-57. 2. Brown, K.E. and Lea, J.F.: "Nodal Systems Analysis of Oil and Gas

Wells," JPT(Oct. 1985) 1751-63. 3. Cheng, M.C.A.: "Perforating Damage and Shot Density Analyzed,"

Oil & Gas J. (March 4, 185) 112-15. 4. Myers, B.W., Clinton, L., and Carlson, N.R.: "Productivity Analy-

sis Tells Where, How Much to Perforate," World Oil (July 1985) 71-75. 5. Mach, J., Proano, E.A., and Brown, K.E.: "Application of Produc-

tion Systems Analysis to Determine Completion Sensitivity on Gas Well Production," paper 81-Pet-13 presented at the 1981 ASME Energy-Sources Technology Conference and Exhibition, Houston, Jan. 18-22.

6. Eickmeier, J .R.: "How to Accurately Predict Future Well Productivi-ties," World Oil (May 1968) 99-106.

613

7. Beggs, H.D. and Brill, J.P.: "A Study of Two-Phase Flow in Inclined Pipes," JPT(May 1973) 607-14; Trans., AIME, 255.

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9. Flanigan, 0.: "Effect of Uphill Flow on Pressure Drop in Design of Two-Phase Gathering Systems," Oil & Gas J. (March 10, 1958).

10. Eaton, B.A. et al.: "The Prediction of Flow Pattern, Liquid Holdup and Pressure Losses Occurring During Continuous Two-Phase Flow in Horizontal Pipelines," JPT(June 1%7) 815-28; Trans., AIME, 240.

11. Duns, H. Jr. and Ros, N.C.J.: "Vertical Flow of Gas and Liquid Mix-tures in Wells," Proc., Sixth World Pet. Cong., Frankfurt-on-Main (1963) 451.

12. Orkiszewski, J.: "Predicting Two-Phase Pressure Drops in Vertical Pipes," JPT (June 1967) 829-38; Trans., AIME, 240.

13. Hagedorn, A.R. and Brown, K.E.: "Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits," JPT (April 1%5) 475-84; Trans., AIME, 234.

14. Cullender, M.H. and Smith, R.V.: "Practical Solution of' Gas-Flow Equations for Wells and Pipelines with Large Temperature Gradients," JPT (Dec. 1956) 281-87; Trans., AIME, 207.

15. Vogel, J.V.: "Inflow Performance Relationships for Solution-Gas Drive Wells," JPT (Jan. 1968) 83-92; Trans., AIME, 243.

16. Standing, M.B.: "Inflow Performance Relationships for Damaged Wells Producing by Solution-Gas Drive," JPT (Nov. 1970) 1399-1400.

17. Fetkovich, M.J.: "The Isochronal Testing of Oil Wells," paper SPE 4529 presented at the 1973 SPE Annual Meeting, Las Vegas, Sept. 30-Oct.3.

18. McGuire, W.J. and Sikora, V.J.: "The Effect of Vertical Fractures on Well Productivity," JPT(Oct. 1960) 72-74; Trans., AIME, 219.

19. Soliman, M.Y.: "Modifications to Production Increase Calculations for a Hydraulically Fractured Well," JPT (Jan. 1983) 170-72.

614

20. Meng, H. et aI.: "Production Systems Analysis of Vertically Fractured Wells," paper SPE 10842 presented at the 1982 SPEIDOE Unconven-tional Gas Recovery Symposium, Pittsburgh, May 16-18.

21. Agarwal, R.G, Carter, R.D., and Pollock, C.B.: "Evaluation and Pre-diction of Performance of Low-Permeability Gas Wells StiplUlated by Massive Hydraulic Fracturing," JPT(March 1979) 362-72; Trans., AIME,267.

22. Cinco-L., H., Samaniego-V., F., and Dominquez-A., N.: "Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Frac-ture," SPEJ (Aug. 1978) 253-64.

23. Soliman, M.Y., Venditto, J.J., and Slusher, G.L.: "Evaluating Frac-tured Well Performance Using Type Curves," paper SPE 12598 present-ed at the 1984 SPE Permian Basin Oil and Gas Recovery Conference, Midland, March 8-9.

24. Gringarten, A.C., Ramey, H.J. Jr., and Raghavan, R.: "Applied Pres-sure Analysis for Fractured Wells," JPT(July 1975) 887-92; Trans., AIME,259.

25. Al-Hussainy, R., Ramey, H.J. Jr., and Crawford, P.B.: "The Flow of Real Gases Through Porous Media," JPT (May 1966) 624-36; Trans., AIME, 237.

51 Metric Conversion Factors bbl x 1.589873 E-Ol

ft x 3.048* E-01 ft3 x 2.831 685 E-02 psi x 6.894757 E+OO

'Conversion factor is exact. SPEPE

Original SPE manuscript received for review Sept. 30, 1986. Paper accepted for publics-tion Dec. 10, 1987. Revised manuscript received Feb. 9, 1988. Paper (SPE 15931) first presented at the 1986 SPE Eastern Regional Meeting held in Columbus, OH, Nov. 12-14.

SPE Production Engineering, November 1988