JERT-16-1235, Liu, 1

The Experimental and Model Study on Variable

Mass Flow for Horizontal Wells with Perforated

Completion

Wei Jianguang Lin Xuesong Liu Xuemei Ma Yuanyuan

Northeast Petroleum University, Daqing City of Heilongjiang Province, 163318

Abstract：The variable mass flow in perforated horizontal wells is very complex. One reason is that the

perforation can increase the roughness of the pipe wall which will increase the frictional pressure drop. The

other is the fluid boundary layer and velocity profile of axial flow will be changed due to the “mixing” of the

inflow with the axial flow. The influences of the perforation parameters and flux rate on the pressure drawdown

in horizontal wellbore are investigated. The perforation parameters include perforation phasing, perforation

diameter, perforation density. According to the experiment results, some modes such as friction factor

calculation model (the accuracy of the model is 4%), “mixing” pressure drop calculation model (the accuracy of

the model is 3%) and total pressure drop calculation model (the accuracy of the model is 2%) are developed.

Key Words: Pressure drop Complex flow Horizontal wellbore Perforated completion

0 Introduction

The affecting rules of the pressure drop for complex flow in horizontal wells are important to the prediction

of production dynamic, the design of borehole trajectory, the optimization of completion parameters and the

inflow control method determination. The variable mass flow in perforated horizontal wellbore is complex

compared with the conventional pipe flow which embodies in two aspects. One is that the pipe wall has bigger

roughness due to the perforations which increase the frictional pressure losses. The other is that the fluid

boundary layer and velocity profile of axial flow changes due to the “mixing” of the inflow with the axial flow. A

considerable number of experimental work on pressure losses of single flow in perforated completed horizontal

wells has been published by many scholars, such as Asheim[1] and Su[2-3] of NTNU, Japanese scholar

Ihara[4-6] ,Yuan[7-10] of Tulsa, OuYang[11-13] of Stanford, Zhou Shengtian[14-15] of CUP(East China),Wang Zhiming[16-17]

of CUP(Beijing), Abdulwahid,M.A.[18]of Andhra University(India), Quan Zhang[19] of CUP (Beijing), Weipeng Jiang[20]

of Tulsa. There are three shortages of the previous work. One is the experiment systems use small size pipes

which cannot satisfy geometric similarity, kinematic similarity and dynamic similarity simultaneously which

induce the results deviated from the actual production. One is that the pipes are made of organic glass instead of

the metal and the fluid is water. The third is that the influences of the perforation phasing, perforation diameter,

perforation density on the pressure drop have not been obtained. In this paper, a full size perforated casing pipe

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with 5.5 inch external diameter is used to simulate the actual production. Three values of perforation phasing,

perforation diameter and perforation density are considered respectively. The Re of axial flow is 1000-20000. The

flux ratio (the ratio of the radial volume inflow at unit wellbore length to the axial volume flow in production pipe)

is 0.01%-10%. The parameters affecting the pressure drop such as perforation phasing, perforation diameter and

perforation density, flux ratio are considered in this paper. The influence law obtained in this paper can offer a

base for development of the pressure drop model.

1 Experiment Introduction

(1) The experiment system include three units：simulation unit, fluid supply and control unit, data collection and

analyzing unit which are shown in Fig.1.

(2) The simulating unit adopts a casing with 124.0 mm inner diameter (external diameter is 5.5 inch)and a casing

pipe with 149.1 mm inner diameter(external diameter is 7.5 inch). This unit is 6.5m long and the distance

between two pressure monitoring nodes is 6 m. To improve the accuracy of the pressure difference transmitter,

the pressure monitoring nodes are connected with difference pressure transmitter by softy sheer plastic pipes.

The accuracy of the pressure difference transmitter is the order of magnitude of 1Pa. The outside wall of the

inner pipe is covered by compact gauze. At the ends of this unit, there are two liquid inlets to ensure the inflow

profile uniform. The simulating unit is shown in Fig.2.

(3) The mineral oil of 10 mPa.s is used instead of crude oil. Before every experiment, the indoor temperature and

viscosity of the mineral oil is measured to make sure the temperature and viscosity in each experiment is same.

(4) Three values of screw perforating phasing in simulation unit are considered： 45°,90°,180°,as shown in

Fig.3. The values of perforation density are 8 meter-1, 16 meter-1, 24 meter-1. The values of perforation diameter

are 10mm, 20mm, 30mm.

(5) The Re of axial flow is 1000-20000. The flux ratio is 0.01%-10%.

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Fig 1 Experiment System for Complex Flow in Horizontal Wellbore

Fig 2 Simulation Unit Diagram

Fig 3A 45°Screw Perforating Phasing

A

A

B

B

C

C

D

D

E

E

F

F

G

G

H

H

A B C D

E F G H

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Fig 3B 90°Screw Perforating Phasing

Fig 3C 180°Screw Perforating Phasing

2 Experiment Results

It is need to emphasize that the inner diameter of these experimental pipes are all 124 mm, the distance of

two pressure measurement nodes is 6m,and the ordinary casing pipe is the casing without perforation. The

relationship curves of Re of axial flow with frictional pressure drop gradient under different perforation density,

diameter, phasing without inflow are shown in fig.4A, fig.4B, fig.4C respectively. From fig.4A, fig.4B, fig.4C, the

effects of perforation density, diameter, phasing on frictional pressure losses are significant. The frictional

pressure losses always increase with the increase of perforation density, diameter and phasing. When the

perforation diameter is 20mm, the perforation phasing is 90°, the Re of axial flow is 20000,the frictional

pressure drop gradient is 376 Pa/m,414 Pa/m,459 Pa/m which is bigger than that of the ordinary casing pipe by

11.11%,22.41%,35.71% corresponding to the perforation density of 8 m-1,16 m-1,24 m-1 respectively. When the

perforation density is 16 m-1,the perforation phasing is 90°,the Re of axial flow is 20000,the frictional pressure

drop gradient is 389Pa/m,414Pa/m,441Pa/m which is bigger than that of the ordinary casing pipe by

15.14%,22.41%,30.46% corresponding to the perforation diameter of 10 mm,20 mm,30 mm respectively. When

the perforation density is 16 m-1,the perforation diameter is 20 mm, the Re of axial flow is 20000,the frictional

pressure drop gradient is 399Pa/m、414Pa/m、430Pa/m which is bigger than that of the ordinary casing pipe by

A

A

B

B

C

C

D

D

A B C D

A

A

A B

B

B

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17.96%,22.41%,27.14% according to the perforation phasing of 45°, 90°, 180°respectively. The results

indicate that the perforation can increase the roughness of the pipe wall which increases the frictional pressure

drop.

To analyze the influence of flux ratio on total and “mixing” pressure drop, the relation curves of total

pressure drop with the flux ratio under different perforation density, diameter, phasing when the Re at outlet

keeps 5000 are obtained in fig.5A, fig.5B and fig.5C, and the relationship curves when the Re of outlet keeps

5000 are obtained in fig.6A, fig.6B and fig.6C. From the fig.5A, fig.5B and fig.5C, we can see that the effect of flux

ratio on total pressure drop is significant. The total pressure drop increases with the flux ratio. The pressure drop

gradient is 28Pa/m、36 Pa/m,45 Pa/m,82 Pa/m which is bigger than the frictional pressure drop of the ordinary

casing pipe(about 29 Pa/m) by -3%,25%,56% ,185% when the flux ratio is 0.01%,0.1%,1%,10% respectively, that

indicates that when the flux ratio is small, the inflow can reduce the frictional pressure drop compared with the

ordinary casing pipe without perforations while the high flux ratio can increase the “mixing” and acceleration

pressure drawdown obviously. The effect of flux ratio is greater than the effect of perforation parameters. From

the fig.6A, fig.6B, fig.6C, the “mixing” pressure drop increases with the flux ratio, but there exists a critical value

of flux ratio. When the actual flux ratio less than the critical value, the “mixing” pressure drop is negative which

means that the inflow fluid can reduce the total pressure drop. When the actual flux ratio bigger than the critical

value, the “mixing” pressure drop is positive and could increases the total pressure drop. The critical flux ratio

increases with the perforation density and perforation diameter. In this paper, the critical value is 0.05%-0.1%.

Fig.7A and fig.7B present the relation curves of the flux ratio with the pressure drop gradient when the Re at

outlet is 5000 and 15000 respectively. From fig.7A and fig.7B, no matter the value of the Re, if the flux ratio is

less than 0.1% the acceleration pressure drop can be neglected. If the flux ratio is greater than 0.1% the

acceleration pressure drop increases significantly with the flux ratio. The Re of axial flow can also affect

acceleration pressure drop. When the Re of axial flow is 5000, the frictional pressure drop gradient is 36 Pa/m,

the acceleration pressure drop is 0.36 Pa/m, 3.6 Pa/m, 34 Pa/m corresponding to the 0.1%, 1%, 10% flux ratio

where the acceleration pressure drop of 10% flux ratio is almost same with frictional pressure drop. When the Re

of axial flow is 15000 and the frictional pressure drop gradient is 242 Pa/m, the acceleration pressure drop is 3.26

Pa/m, 32 Pa/m, 309 Pa/m corresponding to the 0.1%, 1%, 10% flux ratio where the acceleration pressure drop at

10% flux ratio is more than frictional pressure drop.

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Fig 4A Frictional Pressure Drop Gradient versus Re with Different Perforation Density without Inflow

Fig 4B Frictional Pressure Drop Gradient versus Re with Different Perforation Diameter without Inflow

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Fig 4C Frictional Pressure Drop Gradient versus Re with Different Perforation Phasing without Inflow

Fig 5A Total Pressure Drop Gradient versus Flux Ratio with different perforation density with Inflow

Fig 5B Total Pressure Drop Gradient versus Flux ratio with different perforation diameter with Inflow

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Fig 5C Total Pressure Drop Gradient versus Flux Ratio with different perforation phasing with Inflow

Fig 6A “mixing” Pressure Drop Gradient versus Flux ratio with different perforation density with Inflow

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Fig 6B “mixing” Pressure Drop Gradient versus Flux Ratio with different perforation diameter with Inflow

Fig 6C “mixing” Pressure Drop Gradient versus Flux ratio with different perforation phasing with Inflow

Fig 7A The Pressure Drop Gradient versus Flux ratio with Re=5000 of Axial Flow

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Fig 7B The Pressure Drop Gradient versus Flux ratio with Re=15000 of Axial Flow

3 Models

The total pressure drop(the sum of frictional pressure drop, “mixing” pressure drop, acceleration pressure

drop) in horizontal wellbore includes frictional pressure drop, “mixing” pressure drop, acceleration pressure drop

[21]:

w a m tp p p p (1)

The frictional pressure drop is Darcy–Weisbach[22] equation：

2

w w2

L Vp f

D

(2)

The friction factor [3] of perforated wall is

*

w w

8 Re 82.5ln 3.75

2

uB

f f u

(3)

in which B is a function of frictional coefficient 0f of ordinary pipe:

0 0

8 Re 82.5ln 3.75

2B

f f

(4)

and the 0f can be calculated by the Haaland equation[23] given by

1.11

0

1 6.91.8log

Re 3.7

D

f

(5)

The roughness function */u u ( the function of perforation diameter, perforation density and wellbore

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diameter) can be calculated from the empirical correlation[24]:

p p

1*

2

AA

d nu

u D

(6)

From the experiment results, the values of1 2,A A under different perforation parameters are given in table 1.

The fig.8 presents the accuracy of the frictional pressure drop model. From the fig.8 we can see that the relative

error of the model is less than 3%.

Fig.8 Precision Verification of Frictional Pressure Drop Calculation Model

According to the relation curves of the flux ratio with the “mixing” pressure drop(The pressure drop caused

by heat loss and disturbance after the wall entering current and shaft main current’s mixing), the “mixing”

pressure drop is in a linear relationship with flux ratio，so the function is given by

m 1 w 2B lg Bp R (7)

in which the 1 2,B B is given in table 1. The fig.9 presents the accuracy of the “mixing” pressure drop model.

From the fig.9 we can see that the relative error of the model is less than 5%.

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Fig.9A Precision Verification of “mixing” Pressure Drop Calculation Model with Re=5000 in Axial Flow

Fig.9B Precision Verification of “mixing” Pressure Drop Calculation Model with Re=15000 of Axial Flow

Based on the principle of momentum conservation, the acceleration pressure drop is calculated by Eqs.8

2 2

a 2 1p V V (8)

The fig.10 presents the accuracy of the total pressure drop model. From the fig.10 we can see that the

relative error of the model is less than 4%.

+5%

-5%

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Fig.10A Precision Verification of Total Pressure Drop Calculation Model with Re=5000 of Axial Flow

Fig.10B Precision Verification of Total Pressure Drop Calculation Model with Re=15000 of Axial Flow

Tab.1 The Coefficient of Models

Perforation

Density

（shot/m）

Perforation

Diameter

（mm）

Perforation

Phasing

（°）

A1 A2 B1 B2

8 20 90 6.21 12.4

6.66

9.28

16 20 90 8.00 16.0 7.30

24 20 90 9.30 18.6 6.15

16 10 90 5.82 16.0 8.37

16 30 90 9.14 16.0 6.53

16 20 45 6.52 16.0 7.89

16 20 180 9.51 16.0 6.81

+4%

-4%

+4%

-4%

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4 Conclusions

(1) Perforations can increase the roughness of the pipe wall obviously which lead to the additional frictional

pressure drop. The frictional pressure drop increases with the perforation density, diameter, phasing. In this

paper, the perforation density varies 8-24 shots per meter, the perforation diameter is 10-30 mm, the perforation

phasing is 45-180 degree, and the frictional pressure drop is greater than that of ordinary pipe by 11%-35%.

(2) The effect of flux ratio on total pressure drop is significant .The total pressure drop increases with the flux

ratio. The pressure drop gradient is bigger than the frictional pressure drop of the ordinary casing pipe(about 29

Pa/m) by -3%,25%,56% ,185% when the flux ratio is 0.01%,0.1%,1%,10% respectively.

(3) There is a critical value of flux ratio（in this paper, it is 0.05%-0.1%）for the “mixing” pressure drop. When the

actual flux ratio is less than the critical value, the “mixing” pressure drop is negative. When the actual flux ratio is

bigger than the critical value, the “mixing” pressure drop will increase to a positive value. The scale of the flux

ratio increases with the perforation density and perforation diameter.

(4) When the flux ratio is less than 0.1%, the proportion of acceleration pressure drop can be neglected. When

the flux ratio greater than 0.1%, the frictional pressure drop increases with the flux ratio obviously. When the Re

of axial flow is 5000, the acceleration pressure drop is same with the frictional pressure drop under the 10% flux

ratio. When the Re of axial flow is 15000, the acceleration pressure drop is bigger than the frictional pressure

drop by 10% under the 10% flux ratio.

(5) With the increase of flux ratio, the proportion of frictional pressure drop reduces. When the flux ratio is less

than 0.1%, the proportion of frictional pressure drop is 97%.When the flux ratio increases to 1%, the proportion

of frictional pressure drop reduces to 75%.When the flux ration is 10%, the proportion of frictional pressure drop

is as low as 40%.

(6) With the increase of flux ratio, the proportion of acceleration pressure drop increases. When the flux ratio is

less than 0.1%, the proportion of acceleration pressure drop can be neglected. When the flux ratio increases to

1%, the proportion of acceleration pressure drop is about 10%. When the flux ration is 10%, the proportion of

acceleration pressure drop can increase to 45%.

(7) When the flux ratio is less than 1%, the “mixing” pressure drop increases with the flux ratio. When the flux

ratio is bigger than 1%, the proportion of “mixing” pressure drop is always nearly 15%.

(8) The relative error of the results of frictional pressure drop model, “mixing” pressure drop model and the total

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pressure drop model with the experiment results are less than 3%, 5%, 4% respectively.

Nomenclature

tp —total pressure drop in perforated horizontal wellbore, Pa

wp —frictional pressure drop in perforated horizontal wellbore, Pa

ap —acceleration pressure drop in perforated horizontal wellbore, Pa

mp —“mixing” pressure drop in perforated horizontal wellbore, Pa

wf —friction coefficient of the wall in perforation well, dimensionless

L —wellbore length, m

D —wellbore diameter of the perforation well, m

—density of the fluid, kg/m3

V —velocity of axial flow in wellbore, m/s

1V —velocity at inlet of wellbore, m/s

2V —velocity at outlet of wellbore, m/s

Re —Reynolds number of axial flow, dimensionless

pd —perforation diameter, m

pn —perforation density, shot per meter

of —frictional factor of ordinary pipe, dimensionless

—absolute roughness of pipe wall, m

wR —the average-velocity ratio between the wall inflow and wellbore section current, dimensionless

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Author:. Author introduction: Wei Jianguang, male, associate professor. Research area:

theory and technique of well completion optimization, theory and technique of EOR,

seepage mechanism of unconventional oil and gas. Tel:13634663105, E-mail:

weijianguang@163.com.

Natural Science Foundation of Heilongjiang Province project” research on the change

pattern of working viscosity of polymer solution in the reservoir” (No.D2015008)

Natural Science Foundation of China project” research on seepage mechanism of

gas-water two phase in the shale gas reservoir considering the condition of pollution”

(No.51474070)

Figure captions

Fig 1 Experiment System for Complex Flow in Horizontal Wellbore

Fig 2 Simulation Unit Diagram

Fig 3A 45°Screw Perforating Phasing

Fig 3B 90°Screw Perforating Phasing

Fig 3C 180°Screw Perforating Phasing

Fig 4A Frictional Pressure Drop Gradient versus Re with Different Perforation Density without Inflow

Fig 4B Frictional Pressure Drop Gradient versus Re with Different Perforation Diameter without Inflow

Fig 4C Frictional Pressure Drop Gradient versus Re with Different Perforation Phasing without Inflow

Fig 5A Total Pressure Drop Gradient versus Flux Ratio with different perforation density with Inflow

Fig 5B Total Pressure Drop Gradient versus Flux ratio with different perforation diameter with Inflow

Fig 5C Total Pressure Drop Gradient versus Flux Ratio with different perforation phasing with Inflow

Fig 6A “mixing” Pressure Drop Gradient versus Flux ratio with different perforation density with Inflow

Fig 6B “mixing” Pressure Drop Gradient versus Flux Ratio with different perforation diameter with Inflow

Fig 6C “mixing” Pressure Drop Gradient versus Flux ratio with different perforation phasing with Inflow

Fig 7A The Pressure Drop Gradient versus Flux ratio with Re=5000 of Axial Flow

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Fig 7B The Pressure Drop Gradient versus Flux ratio with Re=15000 of Axial Flow

Fig.8 Precision Verification of Frictional Pressure Drop Calculation Model

Fig.9A Precision Verification of “mixing” Pressure Drop Calculation Model with Re=5000 in Axial Flow

Fig.9B Precision Verification of “mixing” Pressure Drop Calculation Model with Re=15000 of Axial Flow

Fig.10A Precision Verification of Total Pressure Drop Calculation Model with Re=5000 of Axial Flow

Fig.10B Precision Verification of Total Pressure Drop Calculation Model with Re=15000 of Axial Flow

Table captions

Tab.1 The Coefficient of Models