Tension and Expansion Analysis of Pipe-in-Pipe Risers: Part A ... · Tension and Expansion Analysis of Pipe-in-Pipe Risers: Part A, Theoretical Formulation. Kevin Chuanjian Man, Bin

Tension and Expansion Analysis of Pipe-in-Pipe Risers: Part A, Theoretical Formulation

Kevin Chuanjian Man, Bin Yue, Adam Szucs, Ricky Thethi 2H Offshore Inc.

Houston, TX, USA

ABSTRACT

This paper provides a mathematical model for accurate and efficient

calculation of the elongation of each string within a pipe-in-pipe top

tensioned riser system due to gravity, pressure and thermal expansions.

The resulting riser system elongation effect is subsequently derived

considering the interactions among all riser strings. In the case where a

tensioner exists, its nonlinear relationship between tension force and

displacement can be captured by using an iterative calculation method.

Examples show that the approach is efficient, and this mathematical

framework is capable of calculating the riser system tension as well as

distributing tension to each string. With the proposed approach, inner

riser pipe pretension can be determined efficiently considering load

conditions during the life time of the riser system.

KEY WORDS: Riser; Pipe-in-pipe; Pretension; Tensioner; Thermal;

Pressure; Expansion

INTRODUCTION

Pipe-in-pipe Top Tensioned Riser (TTR) systems are widely used in the

offshore oil and gas industry. The main feature of a typical pipe-in-pipe

TTR system is a concentric inner string (tubing) protected by one or

more protective outer strings (casings). All strings are rigidly connected

to each other at the top end of the riser. The bottom ends of casings are

fixed to the subsea wellhead, whereas the bottom end of tubing is fixed

to a mud line tubing hanger or downhole packer. Centralizers are

typically used between the outer casing and the inner pipes.

Insulation materials, in the form of solid, liquid, or gas, exist in the

annulus between strings. Generally, the wall temperature and annulus

pressure are higher for the inner strings than the outer strings. If the

strings were not connected, the free elongation of inner string will be

larger than those of the outer strings. However, due to the existence of

end constraints, pressure and temperature variations result in the

redistribution of tensions between the riser strings.

The complexity of the problem is further enhanced by the inner riser

pretension and the riser external forces. The external forces include

weight, environmental loads, and tensioner load. Due to the importance

and complexity of the subject, it is desirable to have a systematic and

accurate methodology to calculate the tension distribution across all

riser strings.

This paper develops a mathematical model for accurate calculation of

the free elongation of each string due to gravity, pressure and thermal

expansions of a pipe-in-pipe TTR system. The resulting riser system

elongation effect is subsequently derived considering the interactions of

all strings. Static calculations are made assuming the risers are

suspended vertically from a floating structure which is assumed as

fixed.

During the development the theoretical formulation, the calculation

considers the effects of:

Riser temperature;

Riser pressure (including end-cap, Poisson effects);

Riser weight and stiffness;

Tensioner system stiffness.

Each riser string is treated as a spring with its stiffness defined by its

length, cross-sectional area, and Young’s modulus. The riser strings are

concentric and connected at both top and bottom ends. The bottom end

is fixed and the top end free to stroke relative to the vessel. The tension

force at the top end, where the tension ring connects the riser to the

tensioner system, varies as a function of tensioner stroke relative to the

vessel, and is considered in the tension distribution calculation.

Below, equations are derived to provide a mathematical model for

tension distribution calculation. The free expansion equations are

provided first; then the equations for riser string expansion with

considering tensioner stiffness are derived; finally, the tension re-

distribution equations are derived. A generic application of the derived

model is then provided as example. The example is provided to show

Learn more at www.2hoffshore.com

how the mathematical model provided in this paper is applied. The

example also emphasizes the importance of the inner pretension for

TTR system. Finally, conclusions are provided.

The comparison between the mathematical model derived in this paper

and finite element analysis is presented in another paper: “Tension and

Expansion Analysis of Pipe-in-Pipe Risers: Part B, Finite Element

Modeling.”

RISER EXPANSION AND TENSION REDISTRIBUTION

EQUATIONS

The expansion and tension distribution equations are presented in this

section. Firstly, the free expansion of a single-string riser system is

analyzed with consideration to the thermal and pressure effect. The

tensioner stiffness is then added into the equations. The tension

redistribution equations of the riser strings are subsequently derived

considering free riser expansion, riser stiffness, and tensioner load.

Riser Free Expansion due to Thermal Effect

Changes of riser string temperature from the undisturbed installed riser

condition result in thermal loads at the surface wellhead. This induces

an expansion in the individual riser string, which is limited by the

mudline, surface wellhead boundary conditions and the overall stiffness

of the combined riser system. Hence, riser effective tension is

redistributed between the strings to compensate for these restraints on

free end thermal expansion. Referring to a text book, e.g. (Schaffer et

al., 1995), equations defining the free thermal expansion due to

temperature variations are given as follows:

OOOTo lTkdlTkL (1)

IIITi lTkdlTkL (2)

Where,

k = pipe coefficient of thermal expansion

ToL = outer riser free thermal expansion due to temperature

variation from undisturbed installed condition over outer

casing string length Ol

TiL = inner riser free thermal expansion due to temperature

variation from undisturbed installed condition over inner

casing string length Il

OT = change in temperature from undisturbed installed

condition (i.e. before production startup) over outer casing

string length Ol

IT = change in temperature from undisturbed installed

condition (i.e. before production startup) over inner casing

string length Il

Riser Free Expansion due to Pressure Effect

Internal fluid pressure and external fluid pressure induce strains in the

individual riser strings. The expansion of a pipe due to internal pressure

is not simply the extension due to the end cap force. Total riser

extension due to pressure is the combined effects of both pressure end

cap and Poisson (ballooning) effects. Referring to API 2RD and using

Hook’s law, equations defining expansion due to pressure in a riser are

given as follows,

EA

LAPAPL ooii

pz

)( (3)

E

LP

t

DPPvL i

ooip

2)(

(4)

L

EDD

DPDPvL

oi

ooiipr

)( (5)

Where,

pzL = riser free expansion due to axial stress

pL = riser free expansion due to hoop stress

prL = riser free expansion due to radial stress

iP = riser mean internal pressure

oP = riser mean external pressure

oD = riser outside diameter

iD = riser inside diameter

L = riser length

E =Young’s Modulus

v = Poisson’s Ratio (typically 0.3 for steel)

A = cross-sectional area of riser steel = )(4

22

io DD

2

4oo DA

2

4ii DA

Eq. 3, 4, 5 are used to calculate riser expansion in axial direction due to

end cap pressure, hoop stress, and radial stress, respectively. The axial

expansions of hoop stress and radial stress represent Poisson’s effect

and can be combined together as follows,

EA

LAPAPvLLL ooii

prppp

)(2

(6)

Eq. (3) and Eq. (6) can be combined to obtain the riser expansion

equation due to pressure,

EA

LAPAPvL ooii

p

)(21

(7)

Eq. (7) can be used for both inner riser pipe and outer riser pipe to

obtain the riser expansion due to pressure. A riser pipe is divided into

several sections to account for variation of either external or internal

pressures. Eq. (7) can also be derived from Lamé equations

(Timoshenko, 1976). Spark derived the strain equation in 1984 starting

from Lamé equations with zero effective stress, which also proves Eq.

(7).

Riser Free Expansion due to External Loads

External loads on the riser system considered in the calculation are the

following:

Tensioner load applied to riser system

Inner casing riser string apparent weight during lock off to the


hanger on the surface wellhead

Tubing or work/drill string apparent weights from surface

BOP to reservoir during hung-off conditions

Addition of BOP, tree, and tensioner system related weight

above tension ring before running tubing

For a regular pipe with uniform size, the expansion under top tension

and weight is given as follows,

LEA

mgLFLF

)5.0( (8)

Where,

F = riser top tension

m = riser mass per unit length g = gravity acceleration constant

Tensioner System

Either a tensioner system or a buoyancy tank can be used to tension a

TTR system. The tensioner system has non-linear force versus stroke.

For most tensioner systems, the relationship between stiffness and

stroke is also non-linear.

Displacement Changes of Combined Riser String

There are many steps involved in riser system installation. The outer

riser pipe is installed first with the rig hook. After the tensioner system

is engaged to the outer riser, the inner riser is run using the rig hook and

engaged at the internal tieback connector (ITBC). The inner riser is then

pretensioned, and hung on the inner riser casing hanger. The pretension

process occurs during riser installation and defines the tension

distribution between riser strings and stroke for all riser conditions

during service. The tensioner stiffness varies during the installation

process as the tensioner strokes due to weight and pressure changes. A

series of combined riser stretch equations are provided in this section

without and with considering the tensioner stiffness.

Let olT be top tension of outer casing and ilT top tension of inner riser

during lock-off the inner riser to the surface wellhead. Then the total

top tension on the tensioner system during lock-off is,

ilollock TTT (9)

The combined riser extension during lock-off is given as,

io

iolockriserl

kk

gLmmTL

)(5.0 (10)

Where,

ok = axial stiffness of the outer riser

ik = axial stiffness of the inner riser

om = outer riser mass per unit length

im = inner riser mass per unit length

Using Eq. 8, the expansion of the outer and inner casings during lock-

off are given as follows,

o

oolol

k

gLmTL

5.0 (11)

i

iilil

k

gLmTL

5.0 (12)

The stretch value of the outer pipe, given in Eq. (11), is different to the

expansion of the riser system, given in Eq. (10). The difference between

these two equations must be considered in order to accurately calculate

the riser stroke.

The top tensions of outer and inner riser during installation affect riser

tension distribution for operations. The pretension process during riser

installation optimizes the riser tension distribution during production.

Pretension is defined as the over-pull on the inner riser above its

submerged weight during installation. API does not directly define

criteria to determine the pretension. However, in order to optimize the

riser response, the pretension is selected such that the base tensions in

primary riser conditions are balanced between the riser strings, and

compression is minimized.

The riser stretch due to pressure and thermal effects with considering

tensioner stiffness is given as follows,

tenio

iPioPoriserP

kkk

kLkLL

(13)

Where,

riserPL = riser expansion due to pressure with tensioner

PoL =outer riser free expansion due to pressure using Eq. 7

PiL =inner riser free expansion due to pressure using Eq. 7

tenio

iTioToriserT

kkk

kLkLL

(14)

Where,

riserTL = riser expansion due to thermal with tensioner

ToL =outer riser free expansion due to thermal using Eq. 1

TiL =inner riser free expansion due to thermal using Eq. 2

Let tensT be top tension of the riser during normal operating condition.

Similar to Eq. 10, the riser extension due to tensioner load is given as,

io

iotensriserNor

kk

gLmmTL

)(5.0 (15)

Where,

riserNorL = riser expansion due to tensioner load

Riser Tension Redistribution Formula

The stretches of riser strings are limited by the subsea wellhead, surface

wellhead and the overall stiffness of the combined riser system. Internal

and external fluid pressure, density, temperature induce strains in the

individual riser strings, which are also limited by the subsea wellhead

and surface wellhead boundary conditions and the overall stiffness of

the combined riser system. Hence, riser effective tension is redistributed

between the strings to compensate for these effects and boundary

conditions. Equations defining changes in effective tension during the

displacement of fluids in the riser are given as follows,


oo

oi

i

ii

oi

o

of gLmkk

kgLm

kk

kF

2

1

2

1 (16)

ii

oi

o

oo

oi

i

if gLmkk

kgLm

kk

kF

2

1

2

1 (17)

Eq. (16) and (17) are used for the tension redistribution due to weight

increase for outer riser inner riser, respectively.

Where,

om = outer riser mass increase per unit length

im = inner riser mass increase per unit length

The tension redistribution equations due to pressure effect are given as

follows,

Po

io

iPioPoooP L

kk

kLkLkF (18)

Pi

io

iPioPoiiP L

kk

kLkLkF (19)

Eq. (18) and (19) are used for the tension redistribution due to pressure

effect for outer riser inner riser, respectively.

The tension redistribution equations due to thermal effect are given as

follows,

To

io

iTioToooT L

kk

kLkLkF (20)

Ti

io

iTioToiiT L

kk

kLkLkF (21)

Eq. (20) and (21) are used for the tension redistribution calculations due

to thermal effect for outer riser inner riser, respectively.

The tension redistribution equations due to tensioner load variation are

given as follows,

iotens

io

ooot FFT

kk

kFF

(22)

iotens

io

iiit FFT

kk

kFF

(23)

Where,

oF = outer riser top tension before tensioner load variation

iF = inner riser top tension before tensioner load variation

Eq. (22) and (23) are used for the tension redistribution due to tensioner

load variation for outer riser inner riser, respectively.

The final riser string tension due to internal and external fluid pressure,

density, temperature for outer riser and inner riser are given as follow,

otoToPofOF FFFFF (24)

itiTiPifIF FFFFF (25)

Where,

OFF = final outer riser top tension with pressure, thermal and

weight effect

IFF = final inner riser top tension with pressure, thermal and

weight effect

Eq. (24) and (25) are used for the final riser string tension due to

internal and external fluid pressure, density, temperature for both inner

and outer riser.

Calculation Procedure

Nonlinear equations are used for tensioner system. Therefore, iterations

are required to solve the proposed model in the paper. The calculation

procedure for the model provided in this paper is shown in Figure 1. A

positive tensioner load is first given to start the calculation. Then the

tensioner strokes and tensioner load are obtained. The current calculated

tensioner load is compared to the given tensioner load or previous strep

tension load. The iterations are finished when the difference between

two successive calculations is less than or equal to the convergence

criterion.

To = Known Tensioner load

Tcal = Calculated Tensioner load

Figure 1 – Flowchart for Calculation Procedure


EXAMPLE OF APPLICATION OF RISER EXPANSION AND

TENSION REDISTRIBUTION EQUATIONS

Software

Microsoft Excel with a Macro using Goal Seek function is applied for

the example calculation.

Environmental Data

A summary of the environmental data used in the example is given

Table 1.

Table 1 – Environmental data

Water depth (m) 1000

Sea water density (kg/m^3) 1025

Gravity, g (m/sec2) 9.8

Riser Temperature

Riser pipes are classified as either hot during operating or ambient when

it is not operating based the load conditions. The summer seawater

temperature is used when risers are not operating. Riser string

temperature profiles for the example are shown in Figure 2.

-1200

-1000

-800

-600

-400

-200

0

200

0 20 40 60 80 100

Ele

va

tio

n a

bo

ve

MW

L (

m)

Temperature (degC)

Riser String Temperature Profiles

Inner Riser Outer Casing

Winter seawater Summer Seawater

MWL Mudline

Figure 2 – Riser string temperature profiles

Tensioner System

Two sets of stiffness parameters for the tensioner system are provided.

The first set of data provides a top tension curve for riser installation.

The other set is for riser operation wherein both outer and inner pipes

exist. A fourth order polynomial equation is applied to estimate the

tensioner load vs. tensioner stroke data, which is shown in Figure 3 and

Figure 4, respectively. Second order polynomial equations can provide a

good estimate of the tensioner stiffness vs. tensioner stroke, which are

shown in Figure 5 and Figure 6.

y = 2.2508x4 - 6.223x3 + 18.963x2 - 72.266x + 249.99R² = 1

0

100

200

300

400

500

600

-1.600 -1.200 -0.800 -0.400 0.000 0.400 0.800 1.200

Ten

sion

er L

oad

(te

)

Riser Stroke (m)

Figure 3 – Tensioner load vs riser stroke profile for installation

y = 3.3883x4 - 9.3682x3 + 28.547x2 - 108.79x + 376.34R² = 1

200

400

600

800

-1.600 -1.200 -0.800 -0.400 0.000 0.400 0.800 1.200T

ensi

on

er L

oad

(te

)

Riser Stroke (m)

Figure 4 – Tensioner load vs riser stroke profile for operation

y = 8.2084x2 + 953.2427x - 65741.5540

R² = 0.9994

3.0E+05

7.0E+05

1.1E+06

1.5E+06

1.9E+06

150 200 250 300 350 400 450

Ten

sion

er S

tiff

nes

s (N

/m)

Tensioner Load (te)

Figure 5 – Tensioner load vs tension stiffness profile for installation

y = 5.4526x2 + 953.2427x - 98967.1676R² = 0.9994

4.0E+05

1.0E+06

1.6E+06

2.2E+06

2.8E+06

3.4E+06

200 300 400 500 600 700

Ten

sio

ner

Sti

ffn

ess

(N/m

)

Tensioner Load (te)

Figure 6 – Tensioner load vs tension stiffness profile for operation


Riser Stackup

A summary of critical riser components for a generic production riser is

given in Table 2. The non-scaled stack-up sketch for a riser

configuration is shown in Figure 7. Calculations of the riser expansion

start from base of the taper stress joint and end at top of the tension

joint. The stiffness of outer riser and inner riser is 2.34×106 N/m and

8.44×105 N/m, respectively.

Table 2 – Major riser component summary

Main Deck

Inner Riser

Outer Riser

Tension Joint

Surface Wellhead

Taper Stress Joint

Subsea Wellhead

Conductor and Casing

MWL = 1000 m

TTR Frame Deck

Drill Floor

Production Tree

Surface BOP

Slip Joint

Note: Slip joint and surface BOP are not installed for normal operating condition.

Figure 7 – Riser Stack-up Overview

Riser Internal Fluids

A summary of the riser internal fluid properties is given in Table 3.

Table 3 – Riser internal fluid property summary

Condition

Internal fluids Fluid Density

(kg/m3)

Surface pressure

(MPa)

Inner Pipe Outer Pipe Inner

Pipe Outer Pipe

Inner

Pipe

Outer

Pipe

Production

(Normal early) Brine Gel 1222 1078 10.3 0.7

Production

(Normal Late) Brine Gel 1222 1078 3.4 0.7

Production

Shut in Brine Gel 1222 1078 34.5 0.7

Hurricane

Evacuation Brine Gel 1222 1078 3.4 0.7

Result Summary

The tension distributions of the inner riser and outer riser for two

different inner riser pretensions, no (zero) pretension and 100 te (981

kN) pretension, are presented in Table 4 and Table 5, respectively. If

the inner riser is not pretensioned, the whole length of the inner riser is

in compression during normal operating condition as given in Table 4.

This is due to the high temperature and high pressure in the inner riser.

A pretension must be applied on the inner riser during installation to

optimize the tension distribution between inner and outer riser strings. If

the inner riser is pretensioned to 100 te during installation, all riser

strings remain in tension during normal operating condition as given in

Table 5,. The tensioner loads and strokes are plotted against the

tensioner stiffness curves in Figure 8 to verify the tension and stroke

calculations. All tensions and strokes match their respective tensioner

stiffness curves.

Table 4 – Riser tension distribution and stroke (pretension = 0)

Component Outer Diameter

(mm)

Wall Thickness

(mm)

Elevation from

Mudline (m)

Start End

Outer Tension Joint 395 – 559 30.5 – 112.3 1020 1035

Outer Standard Joints 272 14.5 6 1020

Inner Riser Pipe 140 10.5 6 1035

BOP? NO No Yes Yes

Description Production

(Normal early)

Production

(Normal Late)

Production

Shut in

Hurricane

Evacuation

Outer Riser

Top Tension

kN 2,972 2,952 1,980 1,898

(te) (303) (301) (202) (193)

Outer Riser

Base Tension

kN 1,946 1,926 954 872

(te) (198) (196) (97) (89)

Inner Riser

Top Tension

kN -80 -54 157 267

(te) (-8) (-6) (16) (27)

Inner Riser

Base Tension

kN -395 -378 -157 -57

(te) (-40) (-39) (-16) (-6)

Combined

Top Tension

kN 2,892 2,898 2,137 2,165

(te) (295) (295) (218) (221)

Combined

Base Tension

kN 1,551 1,548 796 815

(te) (158) (158) (81) (83)

Riser Stroke m 0.304 0.297 -0.084 -0.109

(ft) (1.00) (0.98) (-0.28) (-0.36)

Tensioner Rod

Surface Wellhead

Tensioner Ring

Outer Pipe (Tension Joint)

Inner Pipe

Subsea Wellhead

Outer Pipe (LTSJ)

Inner Pipe


Table 5 – Riser tension distribution and stroke (pretension=100 te)

Figure 8 – Riser Tensioner Load Verification

CONCLUSIONS

A mathematical model for accurate calculation of riser stroke and

tension distribution due to weight, pressure and thermal expansions of a

TTR system has been developed. A nonlinear relationship between

tension force and displacement are captured by using an iterative

method. Examples show that the approach is efficient, and this

mathematical framework is capable of calculating the riser system

tension and tension distribution in each string. With the proposed

approach, inner riser (tubing) pretension can be determined efficiently.

Furthermore, the proposed model can be used to create input data for

finite element analysis considering the load conditions during the life

time of the riser system.

REFERENCES

American Petroleum Institute, (2006). Design of Risers for Floating

Production Systems (FPSs) and Tension-Leg Platforms (TLPs),

Recommended Practice 2RD.

Schaffer, J.P., Saxena, A., Antolovich, S.D., Sanders, T.H., and Warner,

S.B. (1995). The Science and Design of Engineering Materials.

RICHARD D. IRWIN, INC.

Spark, C. P. (1984). “The Influence of Tension, Pressure and Weight on

Pipe and Riser Deformations and Stresses,” J Energy Resources

Technology, Vol 106, pp. 46-54.

Timoshenko, S., (1976). Strength of Material, 3rd Edition, Part II,

Kriegas, Huntington, New York.

Yue, B., Man, C. K., Waters, D. (2013). “Tension and Expansion

Analysis of Pipe-in-Pipe Risers: Part B, Finite Element Modeling.”

Proceedings of ISOPE

BOP? NO No Yes Yes

Description Production

(Normal early)

Production

(Normal Late)

Production

Shut in

Hurricane

Evacuation

Outer Riser

Top Tension

kN 2,425 2,405 1,397 1,377

(te) (247) (245) (142) (140)

Outer Riser

Base Tension

kN 1,399 1,379 370 351

(te) (143) (141) (38) (36)

Inner Riser

Top Tension

kN 701 727 1,032 1,058

(te) (71) (74) (105) (108)

Inner Riser

Base Tension

kN 387 403 717 734

(te) (39) (41) (73) (75)

Combined

Top Tension

kN 3,126 3,133 2,428 2,435

(te) (319) (319) (248) (248)

Combined

Base Tension

kN 1,785 1,782 1,087 1,085

(te) (182) (182) (111) (111)

Riser Stroke m 0.064 0.058 -0.327 -0.333

(ft) (0.21) (0.19) (-1.07) (-1.09)


Documents

Tension and Expansion Analysis of Pipe-in-Pipe Risers: Part A ... · Tension and Expansion Analysis of Pipe-in-Pipe Risers: Part A, Theoretical Formulation. Kevin Chuanjian Man, Bin