Chapter 6

W e l l D e l i v e r a b i l i t y

6.1 Introduction

Well deliverability is determined by the combination of well inflow per-formance (see Chapter 3) and wellbore flow performance (see Chapter 4).While the former describes the deliverability of the reservoir, the latterpresents the resistance to flow of production string. This chapter focuseson prediction of achievable gas production rates from gas reservoirs withspecified production string characteristics. The technique of analysis iscalled Nodal analysis (a Schlumburger patent). Calculation examples areillustrated with computer spreadsheets that are provided with this book.

6.2 Nodal Analysis

Fluid properties, such as gas z-factor and gas viscosity, change with thelocation-dependent pressure and temperature in the gas productionsystem. To simulate the fluid flow in the system, it is necessary to "break"the system into discrete nodes that separate system elements (equipmentsections). Fluid properties at the elements are evaluated locally. Thesystem analysis for determination of fluid production rate and pressure ata specified node is called Nodal analysis in petroleum engineering.

Nodal analysis is performed on the principle of pressure continuity, thatis, there is only one unique pressure value at a given node no matterwhether the pressure is evaluated from the performance of upstreamequipment or downstream equipment. The performance curve (pressure-rate relation) of upstream equipment is called inflow performance curve;the performance curve of downstream equipment is called outflow perfor-mance curve. The intersection of the two performance curves defines theoperating point, that is, operating flow rate and pressure, at the specified

node. For the convenience of using pressure data measured normally ateither bottom hole or wellhead, Nodal analysis is usually conducted usingthe bottom hole or wellhead as the solution node.

6.2.1 Analysis with the Bottom Hole Node

When the bottom hole is used as a solution node in Nodal analysis, theinflow performance is the well Inflow Performance Relationship (IPR)and the outflow performance is the Tubing Performance Relationship(TPR), if the tubing shoe is set to the top of the pay zone. Well IPR can beestablished with different methods presented in Chapter 3. TPR can bemodeled with various approaches as discussed in Chapter 4.

Traditionally, Nodal analysis at the bottom hole is carried out by plottingthe IPR and TPR curves and visually finding the solution at the intersec-tion point of the two curves. With modern computer technologies, thesolution can be computed quickly without plotting the curves, althoughthe curves are still plotted for visual verification.

Consider the bottom hole node of a gas well. If the IPR of the well isdefined by

(6.1)

and if the outflow performance relationship of the node (TPR) is defined by

(6.2)

then the operating flow rate qsc and pressure pw* at the bottom hole nodecan be determined graphically by plotting Equation (6.1) andEquation (6.2) and finding the intersection point.

The operating point can also be solved numerically by combiningEquation (6.1) and Equation (6.2). In fact, Equation (6.1) can berearranged as:

(6.3)

(6.4)

Substituting Equation (6.3) into Equation (6.2) yields:

which can be solved with a numerical technique such as the Newton-Raphson iteration for gas flow rate qsc. This computation can be per-formed automatically with the spreadsheet program BottomHole-Nodal.xls. Users need to input parameter values in the Input Data sectionand run Macro Solution to get results.

Example Problem 6.1

Suppose that a vertical well produces 0.71 specific gravity gasthrough a 2 7/8-in tubing set to the top of a gas reservoir at adepth of 10,000 ft. At tubing head, the pressure is 800 psia andthe temperature is 150 0F, the bottom hole temperature is 200 0F.The relative roughness of tubing is about 0.0006. Calculate theexpected gas production rate of the well using the following datafor IPR:

Reservoir pressure: 2,000 psia

IPR model parameter C: 0.1 Mscf/d-psi2n

IPR model parameter n: 0.8

Solution

This example problem is solved with the spreadsheet programBottomHoleNodal.xls. Table 6-1 shows the appearance of thespreadsheet for the data input and result sections. It indicatesthat the expected gas flow rate is 1,478 Mscf/d at a bottom holepressure of 1,050 psia. The inflow and outflow performancecurves plotted in Figure 6-1 confirm this operating point.

Table 6-1 Input Data and Results Given by BottomHoleNodal.xIs3

Instructions: 1) Input your data in the Input Data section; 2) Run Macro Solutionto get results; 3) View results in table and in the Plot graph sheet.

Input DataGas-specific gravity (yg):Tubing inside diameter (D):Tubing relative roughness (e/D):Measured depth at tubing shoe (L):Inclination angle (6):Wellhead pressure (phf):Wellhead temperature (Thf):Bottom hole temperature (T^).Reservoir pressure (p):C-exponent in backpressure IPR model:n-constant in backpressure IPR model:

Solution

qsc (Mscf/d)

0191383574765956

1,1481,3391,5301,7211,8171,8651,8891,913

Operating flow rate =Operating pressure =

IPR

2,0001,9431,8611,7641,6521,5231,3741,2009877034983532500

1,478 Mscrf/d1,050 psia

TPR

1,0201,0211,0231,0261,0311,0371,0441,0521,0621,0731,0781,0811,0831,084

a. This spreadsheet calculates well deliverability with bottom hole node

Click to View Calculation Example

Botto

m Ho

le Pr

essu

re (ps

ia)

IPRTPR

Gas Production Rate (Mscf/d)

Figure 6-1 Nodal analysis for Example Problem 6.1.

6.2.2 Analysis with Wellhead Node

When the wellhead is used as a solution node in Nodal analysis, theinflow performance curve is the Wellhead Performance Relationship(WPR) that is obtained by transforming the IPR to wellhead through TPR.The outflow performance curve is the wellhead Choke Performance Rela-tionship (CPR). Some TPR models are presented in Chapter 4. CPRmodels are discussed in Chapter 5.

Nodal analysis with wellhead being a solution node is carried out by plot-ting the WPR and CPR curves and finding the solution at the intersectionpoint of the two curves. Again, with modern computer technologies, thesolution can be computed quickly without plotting the curves, althoughthe curves are still plotted for verification.

If the IPR of the well is defined by Equation (6.1), and TPR is representedby Equation (6.2), substituting Equation (6.2) into Equation (6.1) gives

(6.5)

which defines a relationship between wellhead pressure phf and gas produc-tion rate qsc, that is WPR. If the CPR is defined by Equation (5.5), that is,

(6.6)

then the operating flow rate qsc and pressure phf at the wellhead node canbe determined graphically by plotting Equation (6.5) and Equation (6.6)and finding the intersection point.

The operating point can also be solved numerically by combiningEquation (6.5) and Equation (6.6). In fact, Equation (6.6) can berearranged as:

(6.7)

Substituting Equation (6.7) into Equation (6.6) gives

which can be solved numerically for gas flow rate qsc. This computationcan be performed automatically with the spreadsheet program Wellhead-Nodal.xls. Users need to input parameter values in the Input Data sectionand run Macro Solution to get results.

Example Problem 6.2

Use the following given data to estimate gas production rate ofthe well:

Gas-specific gravity: 0.71

Tubing inside diameter: 2.259 in

Tubing wall relative roughness: 0.0006

Measured depth at tubing shoe: 10,000 ft

Inclination angle: 0°

Wellhead choke size: 16 1/64 in

Flowline diameter: 2 in

Gas-specific heat ratio: 1.3

Gas viscosity at wellhead: 0.01 cp

Wellhead temperature: 150 0F

Bottom hole temperature: 200 0F

Reservoir pressure: 2,000 psia

C-constant in backpressure IPR model: 0.01 Mscf/dpsi2n

n-exponent in backpressure IPR model: 0.8

Solution:

This example problem is solved with the spreadsheet programWellheadNodal.xls. Table 6-2 and Table 6-3 show the appear-ance of the spreadsheet for the data input and result sections. Itindicates that the expected gas flow rate is 1,478 Mscf/d at abottom hole pressure of 1,050 psia. The inflow and outflow per-formance curves plotted in Figure 6-2 confirm this operatingpoint.

Table 6-2 Input Data and Solution Given by WellheadNodal.xIs*

Instructions: 1) Input your data in the Input Data section; 2) Run Macro Solutionto get results; 3) View results in table and in the Plot graph sheet.

Input Data

Gas-specific gravity (yg):

Tubing inside diameter (D):

Tubing relative roughness (e/D):

Measured depth at tubing shoe (L):

Inclination angle (9):

Wellhead choke size (Dck):

Flowline diameter (Dfi):

Gas-specific heat ratio (k):

Gas viscosity at wellhead (|i):

Wellhead temperature (Thf):

Bottom hole temperature (T^).

Reservoir pressure (p~):

C-constant in backpressure IPR model:

n-exponent in backpressure IPR model:

Solution

a. This spreadsheet calculates well deliverability with wellhead node.

Click to View Calculation Example

Wellhe

ad P

ressu

re (ps

ia)

Gas Production Rate (Mscf/d)

Figure 6-2 Nodal analysis for Example Problem 6.2.

WPRCPR

Table 6-3 Results Section of WellheadNodal.xls

qsc (Mscf/d)

0191383574765956

1,1481,3391,5301,7211,8171,8651,8891,913

Operating flow rate =Operating pressure =

WPR

1,6001,5541,4891,4111,3211,2181,099960789562399282200

11,470 Mscf/d797 psia

CPR

0104207311415518622726830933985

1,0111,0241,037

Click to View Calculation Example

6.3 References

Greene, W. R. "Analyzing the Performance of Gas Wells." Journal ofPetroleum Technology (July 1983): 31-9.

Nind, T. E. W. Principles of Oil Well Production. 2nd Ed. New York:McGraw-Hill, 1981.

Russell, D. G., J. H. Goodrich, G. E. Perry, and J. F. Bruskotter."Methods for Predicting Gas Well Performance." Journal of Petro-leum Technology (January 1966): 50-7.

6.4 Problems

6-1 A vertical well produces 0.75 specific-gravity gas through a2 7/8-in (ID 2.441 in) tubing set to the top of a gas reservoir at adepth of 8,000 ft. Tubing head temperature is 90 0F, and bottomhole temperature is 160 0F. The relative roughness of tubing isabout 0.0006. Calculate the expected gas production rates of thewell at wellhead pressures of 200 psia, 300 psia, 400 psia, 500psia, and 600 psia using the following data for IPR:

Reservoir pressure: 1,800 psia

IPR model parameter C: 0.15 Mscf/d-psi2n

IPR model parameter n: 0.85

6-2 Calculate the expected gas production rates of the welldescribed in Problem 6-1 for a 2.259-in ID tubing.

6-3 Use the following data to calculate expected gas productionrate of the well:

Gas-specific gravity: 0.75

Tubing inside diameter: 2.259 in

Tubing wall relative roughness: 0.0006

Measured depth at tubing shoe: 8,000 ft

Inclination angle: 0°

Wellhead choke size: 24 1/64 in

Flowline diameter: 2 in

Gas-specific heat ratio: 1.3

Gas viscosity at wellhead: 0.01 cp

Wellhead temperature: 120 0F

Bottom hole temperature: 180 0F

Reservoir pressure: 2,000 psia

C-constant in backpressure IPR model: 0.01 Mscf/dpsi2n

n-exponent in backpressure IPR model: 0.8

6-4 Modify spreadsheet program BottomHoleNodal.xls toincorporate the Forchheimer equation for IPR. SolveProblem 6-1 using estimated A and B values from C and nvalues.

6-5 Modify spreadsheet program WellheadNodal.xls toincorporate the subsonic choke flow equation. SolveProblem 6-3 for flow line pressures of 200 psia, 300 psia,400 psia, 500 psia, and 600 psia.