Chapter 3

G a s R e s e r v o i r D e l i v e r a b i l i t y

3.1 Introduction

Gas reservoir deliverability is evaluated using well inflow performancerelationship (IPR). Gas well IPR determines gas production rate as a non-linear function of pressure drawdown (reservoir pressure minus bottomhole pressure). Gas well IPR also depends on flow conditions, that is,transient, steady state, or pseudosteady state flow, which are determinedby reservoir boundary conditions. This chapter presents methods that canbe used for establishing gas well IPR under different flow conditions.Both analytical methods and empirical methods are discussed. Exampleproblems are illustrated and solved using computer programs providedwith this book.

3.2 Analytical Methods

A general solution to pseudosteady state flow in a radial-flow gas reser-voir is expressed as (Economides 1994):

(3.1)

where q is the gas production rate in Mscf/d, k is the effective perme-ability to gas in md, h is the thickness of pay zone in ft, m(p) is the realgas pseudopressure in psi2/cp at the reservoir pressure p in psi, m(pwf) isthe real gas pseudopressure in psi2/cp at the flowing bottom hole pressure

pwf, T is the reservoir temperature in R, yw is the radius of drainage areain ft, Yw *s wellbore radius in ft, s is skin factor, and D is the non-Darcycoefficient in d/Mscf. The skin factor and non-Darcy coefficient can beestimated on the basis of pressure transient analyses.

As the real gas pseudopressure is difficult to evaluate without a computerprogram, approximations to Equation (3.1) are usually used in the naturalgas industry. At pressures lower than 2,000 psia,

(3.2)

where p^ is the base pressure, /u is the average gas viscosity, and z is theaverage gas compressibility factor. Equation (3.1) can then be simplifiedusing a pressure-squared approach such as:

(3.3)

At pressures higher than 3,000 psia, highly compressed gases behave likeliquids. Equation (3.1) can be approximated using pressure approach as:

(3.4)

where Bg is the average formation volume factor of gas in rb/scf.

Example Problem 3.1

A gas well produces 0.65 specific gravity natural gas with N2>

CO2, and H2S of mole fractions 0.1, 0.08, and 0.02, respectively.The well diameter is 7-7/8 inches. It drains gas from a 78-ft thickpay zone in an area of 160 acres. The average reservoir pressure

is 4,613 psia. Reservoir temperature is 180 0F. Assuming a Darcyskin factor of 5 and a non-Darcy coefficient of 0.001 day/Mscf,estimate the deliverability of the gas reservoir under pseu-dosteady state flow condition at a flowing bottom hole pressure of3,000 psia.

Solution

This problem is solved with the spreadsheet program TheoreticalDeliverability.xls. The appearance of the first section of thespreadsheet is shown in Table 3 - 1 . Results are shown inTable 3-2.

Table 3-1 The First Section of Theoretical Deliverability.xls3

Instructions: 1) Go to the Solution section and enter a value for real gaspseudopressure at the flowing bottom hole pressure; 2) Run Macro Solution andview results.

Input Data

Effective permeability to gas:

Pay zone thickness:

Equivalent drainage radius:

Wellbore radius:

Darcy skin factor:

Non-Darcy coefficient:

Reservoir pressure:

Flowing bottom hole pressure:

Temperature:

Gas-specific gravity:

Mole fraction of N2:

Mole fraction of CO2:

Mole fraction of H2S:

0.17 md

78 ft

1,490 ft

0.328 ft

5

0.001 d/Mscf

4,613 psia

3,000 psia

1800F

0.65 1 for air

0.1

0.08

0.02

a. This spreadsheet calculates theoretical gas reservoir deliverability.

Click to View Calculation Example

Table 3-2 Results Given by Theoretical Deliverability.xls

Solution 1) Based on the property table, enter a value for the real gaspseudopressure at pressure 3,000 psia 604,608,770 psi2/cp; 2) Run MacroSolution to get result.

3.3 Empirical Methods

Very often it is difficult and costly to obtain values of all the parametersin equations (3.1), (3.3), and (3.4). Empirical models are therefore moreattractive and widely employed in field applications. Two commonlyused empirical models are the Forchheimer model and backpressuremodel. They take the following forms, respectively:

(3.5)

(3.6)

and

Click to View Calculation Example

where A, 5, C, and n are empirical constants that can be determined basedon test points. The value of n is usually between 0.5 and 1. It is obviousthat a multirate test is required to estimate values of these constants. If twotest points are (^1, pwf[) and (q^ Pwp), expressions of these constants are:

(3.7)

(3.8)

(3.9)

(3.10)

Similar to Equation (3.1), Equation (3.5) and Equation (3.6) can be sim-plified using the pressure-squared approach as follows:

(3.11)

(3.12)

and

Similarly, the constants can be determined with test points as follows:

(3.13)

(3.14)

(3.15)

(3.16)

Example Problem 3.2

A gas well produces 0.65 specific gravity natural gas with N2>

CO2, and H2S of mole fractions 0.1, 0.08, and 0.02, respectively.The average reservoir pressure is 4,505 psia. Reservoir tempera-ture is 180 0F. The well was tested at two flow rates:

Test point 1

Flow rate:

Bottom hole pressure:

Test point 2

Flow rate:

Bottom hole pressure:

1,152 Mscf/d

3,025 psia

1,548 Mscf/d

1,685 psia

Estimate the deliverability of the gas reservoir under a pseu-dosteady state flow condition at a flowing bottom hole pressure of1,050 psia.

Solution

This problem is solved with the spreadsheet program EmpiricalDeliverability.xls. The appearance of the first section of thespreadsheet is shown in Table 3-3. Results are shown inTable 3-4.

Table 3-3 The First Section of Empirical Deliverability.xls3

Instructions: 1) Go to the Solution section and enter values for real gaspseudopressure at the tested and desired flowing bottom hole pressures; 2) RunMacro Solution, and view results.

Input Data

Reservoir pressure:

Test point 1, flow rate:

bottom hole pressure:

Test point 2, flow rate:

bottom hole pressure:

Flowing bottom hole pressure:

Temperature:

Gas-specific gravity:

Mole fraction of N2:

Mole fraction of CO2:

Mole fraction of H2S:

4,505 psia

1,152 Mscf/d

3,025 psia

1,548 Mscf/d

1,685 psia

1,050 psia

1800F

0.65 1 for air

0.1

0.08

0.02

a. This spreadsheet calculates gas reservoir deliverability with empirical models.

Table 3-4 Results Given by Empirical Deliverability.xls

Solution

1) Use Forchheimer equation with real gas pseudopressure:

Enter real gas pseudopressure at pressure

Enter real gas pseudopressure at pressure

Enter real gas pseudopressure at the desiredpressure

Click to View Calculation Example

Click to View Calculation Example

Table 3-4 Results Given by Empirical Deliverability.xls(Continued)

Run Macro Solution to get result.

gives q =

2) Use Forchheimer equation with pressure-squaredapproach:

Run Macro Solution to get result.

3) Use backpressure model with real gas pseudopressure:

gives q =

Table 3-4 Results Given by Empirical Deliverability.xls(Continued)

4) Use backpressure model with pressure-squared approach:

gives q =

3.4 Construction of Inflow Performance Relationship Curve

Once a deliverability equation is established using either a theoretical oran empirical equation, it can be used to construct well IPR curves.

Example Problem 3.3

Construct IPR curve for the well specified in Example Problem3.1 with both pressure and pressure-squared approaches.

Solution

This problem is solved with the spreadsheet program TheoreticalIPR.xls. The appearance of the spreadsheet is shown in Table 3-5and Table 3-6. IPR curves are shown in Figure 3-1.

Table 3-5 Input Data Given by Theoretical IPR.xls*

Instructions: 1) Update input data; 2) Run Macro Solution and view result and plot.

Input Data

Effective permeability to gas:

Pay zone thickness:

Equivalent drainage radius:

Wellbore radius:

Darcy skin factor:

Non-Darcy coefficient:

Reservoir pressure:

Temperature:

The average gas viscosity:

The average gas compressibility factor:

Effective permeability to gas:

0.17 md

78 ft

1,490 ft

0.328 ft

5

0.001 d/Mscf

4,613 psia

1800F

0.022 cp

0.96

78 ft

a. This spreadsheet calculates and plots theoretical gas well IPR curves.

Table 3-6 Solution Given by Theoretical IPR.xls

Pwt (psia)

Solution

Bg = 0.000671281 rb/SCF

15

245

475

704

934

1,164

q (Mscf/d)

p Approach

1,994

1,906

1,817

1,726

1,635

1,543

p2 Approach

1,067

1,064

1,057

1,044

1,026

1,004

Click to View Calculation Example

Click to View Calculation Example

Table 3-6 Solution Given by Theoretical IPR.xls (Continued)

Pwf(Psia)

1,394

1,624

1,854

2,084

2,314

2,544

2,774

3,004

3,234

3,463

3,693

3,923

4,153

4,383

4,613

q (Mscf/d)

p Approach

1,450

1,355

1,259

1,163

1,064

965

864

762

658

553

446

337

227

114

0

p2 Approach

976

943

905

862

814

760

700

635

563

486

403

312

216

111

0

Example Problem 3.4

Construct IPR curve for the well specified in Example Problem3.2 with both Forchheimer and backpressure equations.

Solution

This problem is solved with the spreadsheet program EmpiricalIPR.xls. The appearance of the spreadsheet is shown in Table 3-7and Table 3-8. IPR curves are shown in Figure 3-2.

Flowin

g Bo

ttom

Hole

Pres

sure

(psia)

Flowin

g Bott

om H

ole P

ressu

re (ps

ia)

Pressure ApproachPressure-squared Approach

Gas Production Rate (Mscf/d)

Figure 3-1 IPR curves given by the spreadsheet program Theoretical IPR.xls.

ForchheimerBackpressure

Gas Production Rate (Mscf/d)

Figure 3-2 IPR curves given by the spreadsheet program Empirical IPR.xls.

Table 3-7 Input Data and Solution Given by Empirical IPR.xlsa

Instructions: 1) Update data; 2) Run Macro Solution and view results.

Input Data

Reservoir pressure: 4,505 psia

Test point 1, flow rate: 1,152 Mscf/d

bottom hole pressure 3,025 psia

Test point 2, flow rate: 1,548 Mscf/d

bottom hole pressure 1,685 psia

Solution

a. This spreadsheet calculates and plots well IPR curve with empirical models.

Click to View Calculation Example

Table 3-8 Results Given by Empirical IPR.xls

q (Mscf/d)Pwt (psia)

Forchheimer Backpressure

15 1,704 1,709

239 1,701 1,706

464 1,693 1,698

688 1,679 1,683

913 1,660 1,663

1,137 1,634 1,637

1,362 1,603 1,605

1,586 1,566 1,567

1,811 1,523 1,522

2,035 1,472 1,471

2,260 1,415 1,412

2,484 1,349 1,346

2,709 1,274 1,272

2,933 1,190 1,189

3,158 1,094 1,095

3,382 984 990

3,607 858 871

3,831 712 734

4,056 535 572

4,280 314 368

4,505 0 0

Click to View Calculation Example

3.5 References

Economides, M. J., A. D. Hill, and C. Ehlig-Economides. PetroleumProduction Systems, New Jersey: Prentice Hall PTR, 1994, 74-5.

Lee, J. Well Testing. Dallas: SPE, 1982, 76-85.

3.6 Problems

3-1 Run spreadsheet program Theoretical Deliverability.xls for atypical gas reservoir at various bottom hole pressures. Make a

conclusion regarding the accuracies of the/7- and/^-approaches.

3-2 Run spreadsheet program Empirical Deliverability.xls for atypical gas reservoir at various bottom hole pressures. Make aconclusion regarding the accuracies of the Forchheimer andbackpressure models.

3-3 Run spreadsheet program Theoretical IPR.xls for a typical gasreservoir. Under what condition is the IPR sensitive to theNon-Darcy coefficient?

3-4 On the basis of Forchheimer model, derive an expression forgas flow rate as an explicit function of flowing bottom holepseudopressure.