Impact of Turbulence on Gas Well Deliverability Muhammad Shoaib*

Petroleum & Gas Engineering Department Dawood University of Engineering & Technology , Karachi, Pakistan

*Corresponding Author: muhammad.shoaib@duet.edu.pk

Abstract— The productivity or deliverability of a gas well isstrongly influenced by additional pressure drop around thewellbore resulting from permeability damage, partialpenetration/ completion and non-Darcy flow. Non-Darcy flow ismainly due to combination of high velocity around wellbore(turbulence effect) and convective acceleration of fluid particleswhen passing through pore spaces (inertial effect). Non-Darcyflow or turbulence is something which can’t be removed becauseit occurs in gas reservoirs and hence, it needs to be dealt with.Direct assessment of turbulence can be obtained from multi-ratewell test interpretation. In the current research work, effect ofturbulence on gas well deliverability was analyzed analyticallyand validated through sensitivity analysis on well testingsoftware.

Index Terms— Turbulence, gas well deliverability, non-Darcy flow coefficient, absolute open flow potential, inflow performance relationship

I. INTRODUCTION

Mechanical damage, partial penetration/ completion and non-Darcy flow can create additional pressure drop around the wellbore and reduce the gas well deliverability and potential. Non-Darcy flow plays a key role in reducing productivity of a gas well. High gas velocity around the wellbore results in transition from laminar to turbulent flow. On the other hand, variation in tortuous pore sizes results in convective acceleration of fluid particles when they pass through pore spaces. Turbulence and inertial effects occur in gas reservoirs and mainly contribute to non-Darcy flow. Multi-rate deliverability test methods were developed for evaluating the productivity of a gas well in the last century. They can be categorized into: Flow after Flow test, Isochronal test and Modified Isochronal test.

II. METHODOLOGY TO ANALYZE WELL DELIVERABILITY

TEST

Flow after Flow (FAF) test is the basic well deliverability test method presented by Rawlins & Schellhardt [1]. FAF test consists of successive increasing flow rates and the corresponding pressure response is obtained from bottomhole gauges installed near sand-face as shown in Fig. 1.

Two basic relations/ equations are mostly used to analyze deliverability test:

Fig. 1. Flow after Flow Test Schematic

A. Rawlins and Schellhardt Equation

Rawlins and Schellhardt (1935) presented an empiricalrelation between gas flow rate and pressure which can be expressed as equation 1 [1]:

(1)

Performance coefficient ‘C’ and Exponent ‘ n’ can be estimated from back-pressure plot as shown in Fig. 2.

Fig. 2. Rawlins & Schellhardt Method for finding C & n

B. Laminar-Inertial Turbulent Equation

Houpeurt (1959) defined an analytical equation 2 in termsof pseudo-pressure which consists of Darcy and non-Darcy components [2].

(2)

Where:

Rat

e

Pre

ssu

re

Time

Pressure (bara)

Rate (MMscfd)

6

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DOI: 10.24081/nijesr.2016.1.0002

„A‟ and „B‟ can be used for predicting future deliverability

of well [2]. represents pseudo-pressure drop due to laminar flow while represents pseudo-pressure drop due to inertial-turbulent flow effects [3].

Plotting

gives straight line with a

slope equals to „B‟ and y-intercept equals to „A‟ as shown in Fig. 3.

Fig. 3. Pseudo-Pressure Quadratic Approach for finding A and B

III. EFFECT OF TURBULENCE ON WELL DELIVERABILITY

COEFFICIENTS

An attempt is made to derive a mathematical relation of „A‟ & „B‟ w.r.t non-Darcy flow coefficient „D‟ using Rawlins and Schellhardt Equation and ‘ C’ & ‘ n’ w.r.t ‘ D’ using Laminar Inertial Turbulent Equation. Analytical relations were also validated using simulations on well testing software.

A. Rawlins and Schellhardt Method Coefficients

Rawlins and Schellhardt equation [1] is given by equation 3: (3)

Darcy Equation for gas valid under low pressure conditions (P<2000psi) assuming µZ = constant [4] is given by equation 4:

(4)

Considering sand-face as the outward condition and

assuming cylindrical area result in equation 5:

(5)

Re-arranging equation 5 gives:

(6)

Integrating equation 6 results in:

(7)

Rearranging to get the difference of pressure squared for

steady state condition as shown in equation 8:

(8)

If pseudo state condition is assumed with well in center of

circular area and introducing mechanical and turbulence skin, equation 8 can be re-written as equation 9:

(9)

Another way to write equation 3 is:

(10)

Re-arranging equation 10 gives:

(11)

There is no direct relation of „C‟ and „n‟ w.r.t „D‟. Both

values of „C‟ and „n‟ are affected by changing „D‟ and neither of them is constant. In a simple way, relation can be written as equation 12: (12)

A base synthetic case was considered with actual reservoir and well properties. The scenario was constructed on the basis of actual well test data. Simulations were performed on well testing software and sensitivity analysis shows a general trend of „C‟ and „n‟ w.r.t. „D‟ as shown in Fig. 4.

In Fig. 4, value of exponent „n‟ decreases drastically from value near 1 (laminar flow) to value of 0.5 (turbulent flow) with increase in „D‟. Performance coefficient „C‟ increases with „D‟ during drastic decrease in value of „n‟ but afterwards it starts to decrease.

Slope = B

A

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Fig. 4. Relation of C and n with D

B. Laminar Inertial Turbulent Method Coefficients

Houpeurt (1959) presented an analytical equation 13:

(13)

For pseudo-steady condition with well placed in center of

circular area, analytical equation for radial flow can be written as equation 14:

(14)

Equations 15 and 16 express values of „A‟ and „B‟:

(15)

(16)

Fig. 5. Relation of A and B with D

From equation 15, it follows that „A‟ is independent of „D‟ while equation 16 shows that „B‟ is a linear function of „D‟ with a slope inversely proportional to „kh‟. The behavior is confirmed also by the sensitivity analysis as shown in Fig. 5.

C. Effect of Turbulence on Absolute Open Flow Potential (AOFP)

Sensitivity analysis was performed by changing the values of „D‟. Effect of changing „D‟ was noticed on absolute open flow potential „AOFP‟. According to Fig. 6, „AOFP‟ decreases drastically at low values of „D‟ while decrease in „AOFP‟ becomes small at higher values of „D‟.

Fig. 6. AOFP vs. D

D. Effect of Turbulence on Inflow Performance Relationship (IPR)

Inflow Performance Relationship (IPR) was developed using the simulated base case and then sensitivities of „D‟ were performed. Turbulence decreases the productivity of well so IPR shifts downward with increase in non-Darcy flow coefficient „D‟ as shown in Fig. 7.

Fig. 7. IPR vs. D

IV. CONCLUSIONS

Turbulence in gas wells has a significant impact on gas well deliverability and potential. From few analytical steps, it is possible to see that laminar flow coefficient „A‟ is independent of „D‟ while inertial-turbulent flow coefficient „B‟ is a linear function of „D‟ with a slope inversely proportional to

0.5

0.6

0.7

0.8

0.9

1.0

0

50

100

150

200

250

0 0.0005 0.001 0.0015

n (

-)

C (

Msc

fd/b

arsa

2)

D - (Mscf/day)-1

Rawlins and Schellhardt Method

Performance Coefficient 'C'

Exponent 'n'

0.00

0.02

0.04

0.06

0.08

0.10

0.12

500

520

540

560

580

600

0 0.0005 0.001 0.0015

B -

[psi

2/c

p]/

Msc

fd2

A -

[psi

2/c

p]/

Msc

fd

D - (Mscf/day)-1

Laminar Inertial Turbulent (LIT) method

Laminar Flow Coefficient 'A'

Inertial Turbulent Flow Coefficient 'B'

0

100

200

300

400

500

600

0 0.0005 0.001 0.0015A

OFP

(MM

scfd

)

D - (Mscf/day)-1

Absolute Open Flow Potential

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

Bot

tom

hole

Flo

win

g Pr

essu

re (b

arsa

)

Gas Rate (MMscfd)

D = 0

D = 0.0005

D = 0.001

D = 0.0015

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„kh‟. Conversely, „D‟ is jointly dependent from turbulent exponent „n‟ and performance coefficient „C‟. Performance coefficient „C‟ increases at low values of non-Darcy flow coefficient „D‟ while it starts to decrease when „D‟ increases. Exponent „n‟ decreases as non-Darcy flow coefficient increases with a hyperbolic trend. Inflow Performance Relationship and Absolute Open Flow Potential decrease drastically with increase in turbulence.

ACKNOWLEDGMENT

Authors would like to acknowledge the support of Kappa Engineering for providing the software used in the research.

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