metering of two-phase geothermal wells using pressure pulse technology

Lukasz Piwoda

Metering of Two-Phase Geothermal Wells Using Pressure Pulse Technology

Diploma Thesis Norwegian University of Science and Technology

Department of Petroleum Engineering and Applied Geophysics

July 2003

Acknowledgement __________________________________________________________________________

i

I wish to thank Professor Jon Steinar Gudmundsson for being my supervisor. I am grateful

for his enthusiasm, an ocean of suggestions, and excellent supervision throughout this work.

Thanks for my supervisor in Poland Dr. Ing. Czeslaw Rybicki to recommend me for

“Erasmus Link to Norway” scholarship and his efforts to enable my take on the study at

NTNU. I am thankful to ING AG Leipzig and Norwegian University of Science and

Technology for financing my scholarship. I wish to thank Mr Wolfgang Laschet from Office

of International Relations for his help to organize my stay in Norway. I also want thank to

Professor Danuta Bielewicz and Professor Jan Falkus for their efforts and engagements into

international cooperation between universities, and for their appreciate help to surmount the

official adversity.

In addition I want to thank Jon Rønnevig, Kjell Korsan and Harald Celius from Markland AS,

for their guidance into the computer simulations and suggestions towards the obtained results.

List of Contents __________________________________________________________________________

ii

List of contents Acknowledgement …………………………………………………………………………….i List of contents ………………………………………………………….……………………ii Nomenclature…………….……………………………………………………………………v Abstract……………………………………………………………………………………..…1 Introduction …………….……………………………………………………………………..2 1. Metering of Multiphase Wells 3

1.1 Introduction………………………………………………………………………….....5 1.2 Overview of multiphase metering……………………………………….......................5 1.3 Challenges and accuracy…...……………………………………….…………...…......7 1.4 Metering techniques…...…………………..………………………….……………......9 1.5 MFMs Projects…...………………………………………………….……………......12

2. Pressure Pulse Technology 13

2.1 Pressure Pulse method…...……………………………………...……………...…......14 2.2 Water-hammer effect…...…………………………………………………...…….......14 2.3 Theory and equation…...……………………………………...…………...….............15 2.4 Pressure surge in wellbores…...…………………………………...…………...…......16 2.5 Mass and volume flowrates…………………...............................................................17 2.6 Flow condition analysis……………………………..……………………...................19 2.7 Concluding remarks……………………………............………………......................20

3. Geothermal Applications 27

3.1. Geothermal energy……………………………...…………...………….....................28 3.2 Geothermal well flow……………………………........................................................29 3.3 Well performance ……………………….....................................................................32

4. Multi-phase Flow in Wells 37

4.1. Introduction………………………………………………………………………..…38 4.2. Main difficulties………………………………………………….…………………. 38 4.3. Phase behaviour……………………………………………………………………... 39 4.4. Definition and variables…………………………………………………………….. 42 4.5 Fluid properties…………………………………………………………………....… 44 4.6. Flow patterns………………………………………………………………………... 44 4.7. Pressure gradient……………………………………………………………………..47 4.8. Multiphase flow models……………………………………….……………………..48 4.9. Duns and Ros correlation for multiphase flow in oil wells…………………………..51 4.10. Duns and Ros modifications………………………………………………………..51 4.11. Orkiszewski correlation for multiphase flow in geothermal wells……………..…..52

List of Contents __________________________________________________________________________

iii

5. Sped of Sound in Two-Phase Mixtures 54

5.1 Introduction…………………………………………………………………………… 55 5.2 Compressibility of two-phase mixtures……………………………….……………… 55 5.3 Compressibility of steam-water system……………………………………………… 57 5.4 Acoustic velocity models …………………………………………………………..… 62 5.5 Attenuation mechanisms of sound wave……………………………………………… 66 5.6. Concluding remarks……………………………………………………………….… 68

6. Case studies 74

6.1 Calculation purpose……………………………………………………………………75 6.2 Water-hammer and line packing in oil wells…………………………………..………75 6.3 Water-hammer and line packing in geothermal well…………………………………106

7. Discussion 156

7.1 Multiphase flow correlations………………………………………………...……… 157 7.2 Acoustic velocity profile………………………………………...……………………158 7.3 Line packing………………………………………………………………………… 159 7.4 Size of the pressure pulse…………………………………………………………..…161

8. Conclusions 162 9. References 164 Appendix A – Multiphase Metering Projects 174 Appendix B – Duns and Ros, Orkiszewski - Multiphase Flow Correlations 183

B.1 Duns and Ross Correlation…………………………………………………..…….…184 B.2 Orkiszewski Correlation……………………………………………………………...192

Appendix C – Sound Wave Propagation Process in Steam Water Mixture 195 Appendix D – PipeSim 2000-Multiphase Flow Simulator 198

D.1 PipeSim Well Performance Analyses……………………………………………….199 D.1.1. Fluid Properties Correlations…………………………………………….…..199 D.1.2 Advanced calibration data………………………………………………..…..203

D.2 Profile model………………………………………………………………..………205 D.2.1. Detailed model………………………………………………………………205 D.2.2. Simplified model………………………………………………….…………206

D.3 IPR Data………………………………………………………………………….…207D.4 Matching option……………………………………………………………….……208D.5 VLP correlations and applications……………………………………………….…210

List of Contents __________________________________________________________________________

iv

Appendix E – HOLA 3.1-Multiphase Flow Simulator 213

E.1 Introduction………………………………………………………………………....215 E.2 Governing equations …………………………………………………………….…215 E.3 The computational models of HOLA 3.1……………………………………..……218 E.4 Heat loss parameters…………………………………………………………..……218 E.5 Wellbore geometry…………………………………………………………………219 E.6 Feedzone properties……………………………………………………………...…220 E.7 Velocities of individual phases…………………………………………………..…221 E.8 Productivity Index estimation………………………………………………………222

Appendix F – Simulation results in oil wells 226

F.1 Well A1…………………………………………………………………..…………227 F.2 Well A2………………………………………………………………………..…....234 F.3 Well B………………………………………………………………………………240 F.4 Well C……………………………………………………..………………………..246

Appendix G – Simulation results in geothermal wells 252

G.1 Well D1…………………………………………………………………..………...253 G.2 Well D2…………………………………………………………………..………...256 G.3 Well E1……………………………………………………………………..……....259 G.4 Well E2……………………………………………………………………..……....262 G.5 Well F1………………………………………………………………………..…....265 G.6 Well F2…………………………………………………………………………......268

Nomenclature __________________________________________________________________________

v

a – acoustic velocity

A –cross section area

B – volume factor

Cp – specific heat capacity at constant pressure

CV – specific heat capacity at constant volume

d – diameter

f – friction factor

g - absolute gravity

h – enthalpy

H – liquid holdup

ID – inner diameter

k – permeability

K – slip ratio

KS – isentropic compressibility

KT – isothermal compressibility

L – length

m –mass flow rates

p – pressure

PI – productivity index

q – volumetric flow rates

R – individual gas constant

Re – Reynolds number

RS – solution gas-oil ratio Sm3 gas/ Sm3 oil

S – entropy

t – time

T – temperature

u – velocity

WC - water cut

V – volume

x – mass fraction

z - direction opposite to gravity

Nomenclature __________________________________________________________________________

vi

Greek letters:

α – void fraction

β – water-oil volumetric factor

γ – specific heats ratio

µ – dynamic viscosity

v – specific volume

ρ – density

Abstract __________________________________________________________________________

1

Multiphase flow measurement is of vital importance in petroleum and geothermal industry.

Overview of currently available metering techniques has been made in present work. Pressure

Pulse method is a new developed method which propose a different approach to measure

two-phase flow in wells. The pressure effects after rapid valve closure that built up the

method were illustrated. The inspection of the types of geothermal reservoirs allowed

characterizing typical parameters of high enthalpy geothermal well. The difficulties to predict

the multiphase flow in wells are presented together with description of the definitions and

variables that need to be calculated. Multiphase flow models were examined and two most

appropriate correlations have been selected for oil and geothermal wells. The speed of sound

in two-phase mixtures was calculated. The available models to estimate acoustic velocity

were studied and verified with respect to their limitations. The compressibility of steam-water

system under the well flow conditions, required for calculations was derived from

thermodynamics definitions. The simulations were performed in PipeSim 2000 and HOLA

3.1 programs for oil and geothermal wells respectively, in order to demonstrate the Pressure

Pulse method. The case studies include three different North Sea oil wells and likewise three

typical high enthalpy geothermal wells. Inflow performance and tubing performance

calculations allowed extending the calculation for different diameters and flowrates. The

results are presented in form of the tables and plots. Obtained results for oil and geothermal

cases were compared to each other. All parameters that affect the acceleration pressure

(pressure increase after rapid valve closure) and pressure built up in wells are discussed. The

work ends with conclusions towards the performed calculations and gives the assessment for

possible application of the Pressure Pulse method to meter the flow in two-phase geothermal

wells.

Introduction __________________________________________________________________________

2

Pipe-flow mixtures of crude oil, gas and water are common in petroleum industry, and yet

their measurements nearly always present difficulties. The traditional solution is first to

separate the components of the flow, and then measure the flow rate of each component using

conventional single-phase flow meters. This method is both inconvenient and expensive to

use for well monitoring. In addition the separation is not accurate, about 10% (Millington,

1999). Current multiphase meters have similar accuracy, they employ the complex techniques,

and some of them contain the dangerous radioactive materials as discussed in Chapter 1.

In geothermal wells producing steam and water mixture under various operating conditions

the capability accurately measure the flow is also of value importance for several reasons

similar to petroleum industry. These are general evaluation of the geothermal reservoir under

proper reservoir management, optimalisation of the wellbore design from well deliverability

considerations and minimization of scale deposits in the wellbore (Ragnarsson, 2000).

The background for this thesis work is a new method to measure multiphase follow

(Gudmundsson and Falk, 1999; Gudmundsson and Celius, 1999), developed at Norwegian

University of Science and Technology (NTNU). The metering method is simple, requires

little space, and is cost effective with at least the same accuracy as the competitors

(Gudmundsson and Celius, 1999). The multiphase flow meter is based on measurements of

pressure magnitude and pressure build-up. The output is velocity and density of the gas-

liquid flow.

This thesis work concerns multiphase metering, specifically pressure transients caused by a

rapid valve closure in oil and geothermal wells. Pressure propagation in fluids is closely

related to sound velocity. The acoustic velocity in two phase mixtures varies significantly

from this in single liquid or gaseous phase, and depends on physical properties of every

mixture constituents. Available models for acoustic velocity in two-phase mixtures need to be

verified according to their limitations in order to find the most appropriate for particular

calculations.

Introduction __________________________________________________________________________

3

Multiphase flow is a complex, turbulent and highly nonlinear process, which can not be fully

described mathematically due to increased numbers of flow parameters. Computer

simulations base on semi-empirical correlations were developed in order to predict the

pressure and fluid parameters changes across the wellbore.

Calculating flowing pressure profiles in oil wells, phase transfer between oil and gas requires

a rather simple treatment, and is accomplished trough the use of solution gas-oil ratio – Rs

relationship. In geothermal wells, however phase transfer between water and steam attains

critical importance and calculations must incorporate the steam tables accurately. Pressure

profile calculations for geothermal wells vary from those for oil well in another important

aspect in that the temperature of the fluid must be computed precisely.

This thesis describes how the Pressure Pulse method can be used to meter the flow in high

enthalpy two-phase steam-water geothermal wells similarly to oil wells. The calculations

performed aim to estimate the size of pressure pulse after the valve closure and determine the

parameters affecting the early pressure build-up.

Chapter 1

Metering of Multiphase Wells

1. Metering of Multiphase Wells __________________________________________________________________________

5

1.1 Introduction

Multiphase means a single component existing in a variety of phases such as steam, water

and ice. In the oil industry multiphase refers to a stream of fluid containing a liquid

hydrocarbon phase (crude or condensate), a gaseous phase (natural gas, and non hydrocarbon

gases), a produced water phase, and solids phase (sand, wax, or hydrates). In general the

quantities of solids produced are minimal and thus have less impact than the liquid and gas

phases. In present thesis work some simplification will be made and two phase, liquid phase

and gas phase will be considered. The mixture of two immiscible fluids will be termed as

liquid phase, regardless to components number. The mixture of gases flowing together will

usually, unless there is a large density difference and little turbulence, diffuse together and

can be treated as single homogenous phase (McNeil, 1990).

Multiphase measurement is the measurement of the liquid and gas phases in a production

stream without the benefit of prior separation of the phases before entering the meter.

1.2 Overview of multiphase metering

While two-phase and multiphase flows have been common throughout petroleum industry for

many years, there has until very recently been little or no demand for real-time metering of

such flows. Traditionally the problem was circumvented by separating the flow into its

constituent components, which allowed straightforward single phase metering techniques to

be used (Theuveny et al., 2001). This approach was very practical and effective, but did give

rise to processing systems which were quite inflexible in terms of their capability to handle

fluctuating flowrates, varying water contents, and changes in the physical properties of

production fluids. However in the early years of offshore North Sea production this was not a

major problem, and at that time - pre 1980 - there was little or no impetus to develop more

sophisticated metering technology that could perhaps dispense with separation equipment and

expensive metering facilities (Falcone et al., 2002).


6

During the 1980s the process of gradually declining oil production from the major North Sea

fields started, so in the interests of operational cost effectiveness, there was a move to use

existing platform based process plant for other production roles (Steward, 2003). To maintain

production levels, smaller satellite fields which were previously uneconomic to produce on a

stand alone basis, were tied back to existing platform based infrastructure.

From a technological point of view this introduced a step change in the complexity of

production. There were now numerous fields, typically with quite different oil properties,

water contents and gas fractions, all being produced through process plant designed for the

early years of single-well production (Theuveny et al., 2001). Furthermore, the water

contents and gas fractions started to increase, and this exacerbated the production problems

even further. It began to emerge quite quickly that more operationally flexible multiphase

technologies were going to be needed, if not immediately, certainly within five to ten years

(Steward, 2003). For existing platforms the prime purpose of this new technology would be

to improve processing flexibility, and for new field developments the aim would be to

completely eliminate the need for costly and bulky platform based process plant (Falcone et

al., 2002). The ultimate aim was of course to move towards remote subsea instrumentation.

The key driver at all times being lower production costs through reduced initial capital

expenditure, and reduced operating manpower.

To take up these challenges, the growth in multiphase research and development since the

early 1980s has been exponential, especially with regard to metering, and today there are a

variety of multiphase flowmeter (MFM) installed onshore and offshore. It appears to be no

reduction in new metering developments (Steward, 2003). However the actual growth rate of

installations has been lower than initial industry forecast suggested (Falcone et al., 2002). Oil

companies have been hesitant to invest in expense meters with limited tracks record. Figure

1.1 shows actual trend up, tied with very low level of utilisation MFM technology before the

2000. The reasons may be the fact that when operators decide between a traditional approach

to the production facilities and one including MFM, must compare the capital and operating

expenses of each solution. Very little operational history of MFM cause difficult to predict

the operating costs (Jamieson, 1999). This difficulty results from relatively low number of

MFM applications worldwide, allow claiming that widespread implementation of MFM


7

cannot take place until expertise is spread more widely trough oil industry (Falcone et al.,

2002).

0

100

200

300

400

500

600

700

800

900

Num

ber o

f MFM

ista

latio

ns

Offsore subsea 0 0 0 1 5 22 20 43 55

Offshore topside 1 2 10 17 29 52 66 103 210

Onshore 5 5 3 6 24 65 78 129 542

1992 1993 1994 1995 1996 1997 1998 1999 2000

Figure 1.1 Grow rates of MFM installations (Falcone et al., 2002)

1.3 Challenges and accuracy

The level of difficulties in accurately measuring the multiphase stream is increased

dramatically over single phase measurement. Single phase fluids can be quantified by

knowing about the pressure, fluid density, viscosity, compressibility and geometry of the

measurement device (Williams, 1994). Unfortunately, multiphase fluids do not act in the

same manner as single phase fluids and above variables of each phase would not quantify

multiphase flow.

Multiphase flow is a complex, turbulent, highly non linear process. Williams (1994), and also

King (1999) give the brief description of the processes that may take place as the different

phases flows simultaneously. The phases interact with each other gas may evolve out of the


8

solution, or is absorbed into the liquid, waxes and hydrates may precipitate etc. If the single

component exists in the two phases there is significant mass transfer and thus mixture quality

may be considered variable. The components do not mix homogenously, and tend to remain

separate, the water does not mix well with the oil, and gas remains separate from the liquid

phase. Both phases flow at the different velocities. It is common for gas and liquid to flow at

the different rates. Very complex flow regimes can exist and are dependent on the relative

velocity of the phases, fluid properties, pipe configuration and flow orientation. The

mentioned above and other relevant to the multiphase flow parameters are described in

Chapter 4, which deals with multiphase flow in wells. The parameters definitions and

relationship between them are given together with the correlations developed in order to

predict the flow behaviour.

Expectations of MFM performance in the early days were concluded and sometimes in the

fiscal range of accuracy. Such levels of accuracy were, and never will be achievable by

present technology (Steward, 2003). Over the last ten years, a gradually more realistic

assessment of uncertainty capabilities has evolved. To date, no international regulations for

MFM accuracy has been delivered. Varying level of accuracy requirements exists in

multiphase measurement depend on how the information will be utilized. Essentially, three

main accuracy requirements exist for metering multiphase fluids (Falcone et al., 2002):

- approximately 5 -10% for reservoir management,

- approximately 2-5% for production allocation,

- and approximately 0.25-1% for fiscal metering, are anticipated to be required.

However because of high complexity of multiphase mixtures it may be optimistic to claim

that the above ranges of accuracy apply to any regime and for any chemistry of the fluids.


9

1.4 Metering Techniques

Under the multiphase flow circumstances the following parameters are required to compute

flowrates of each phase:

- the cross-sectional area of the pipe occupied by each phase

- the axial velocity of each phase

- density of each phase.

The cross-section area and phase velocities give the volumetric phase flowrates. The product

of phase densities and phase volumetric flowrates gives the phase mass flow rate.

Unfortunately, at the present time there is no method of measuring phase fraction directly,

they are derived from two independent measurements, coupled with the continuity equation

which requires the sum of oil water and gas phase fraction to equal unity. Typically, two

variables independent are the density of the entire flow, and the water content in the liquid

phase. Once these are measured, some simple mathematical analysis allows the individual

phase fractions to be calculated.

With these technology limitations, projects aimed at developing multiphase meters have

tended to adopt one of two metering strategies (Millington 1999):

- A set of sensors that take volume measurements, which when combined are capable

of isolating the individual phase fractions. A combination of flow models and velocity

measurements are used to derive the phase velocities as functions of time. To determine

densities, temperature and pressure are measured and assumed equal in all phases.

- A set of sensors which again take volume measurements, but which also require flow

to be conditioned such that only one mixture velocity measured is assumed to be required .

The overall mixture density is considered representative of the three individual phases. Phase

fraction data is required as above.


10

Following sensors and techniques are commonly used:

Gamma Densitometers – consist of radioactive source and detector, placed so that the beam

passes trough the flow and is monitored on the opposite side of the multiphase mixture. The

amount of radiation that is absorbed or scattered by the fluid is a function of both fluid

density and energy level of the source. Typical radioactive sources used include isotopes of

caesium, barium or americium. Single energy gamma sensors are those that incorporate only

one source or monitor only one energy level from source. These devices are often used to

measure the density of the multiphase mixture. Dual energy gamma sensors measure the

absorption of two separate energy levels. The two energy levels are provided either by two

isotopes or by a single isotope that has two discernible levels. If two energy levels are far

enough apart, these two independent absorption measurements can be used to determine the

oil, gas, and water phase volume fractions. The densitometers are frequently calibrated by

filling the device with known fluids, typically gas (or empty pipe) and water.

Capacitance Sensors – measure the dielectric properties of fluid. Each sensor consists of a

pair of metal plates or electrodes. These are mounted on the pipe wall or are otherwise

located so that the fluid occupies the space between them. The capacitance of the fluid is

measured by varying the voltage difference between the plates and measuring the resulting

electric current between them. From the capacitance, the dielectric constant of the mixture

can be calculated. Since the dielectric constant of the mixture is a known function of the

composition, this information can be used to calculate the volume fractions of oil, gas, and

water phases. This technique will work for mixtures in which the liquid (oil/water mix) is oil

continuous. Since the water phase is a much better conductor of electricity, water continuous

mixtures will effectively "short" the capacitance plates rendering the measurement ineffective.

For water continuous liquids, an approach based on conductance is used (see below).

Conductance / Inductance Sensors – use an electrical coil around the pipe to induce a current

in the flowing multiphase mixture. The magnitude of this induced current is related to the

dielectric constant of the mixture, which can be used to determine the (mixture composition

as with the capacitance and microwave sensors.


11

Microwave Sensors – measure the dielectric properties to help determine the phase fractions

of the multiphase mixture. The sensor consists of emitters and receivers (antennae) of

electromagnetic waves in the MHz or GHz range (microwaves). The dielectric constant of the

mixture is a function of both the frequency of the waves and the mixture conductivity. The

measured dielectric constant is a volume weighted average of the individual phase dielectric

constants. The conductivity and dielectric constant of the water phase is a function of salinity.

As such, meters that use this technique either need brine salinity as a calibration variable or

have some other way of estimating it on-line.

Cross Correlation Techniques – use two similar measurements, each in a different axial

location in the pipe. By comparing the two measurements, the velocity of the flow feature is

determined, for example, the time required for a bubble to travel between the two sensors.

Implicit in this technique is a measurable amount of non-homogeneity in the multi phase flow.

For this reason, many available meters require the Gas Volume Flow (GVF) to be within

certain limits, far enough from the pure liquid (GVF = 0) and pure gas limits (GVF = l) that

the flow does not appear homogeneous to the sensors. Gamma densitometers, microwave

sensors, and capacitance sensors are used in MPM systems for cross correlation.

Venturi Meters – consist of a gradual restriction in the flow path, followed by a gradual

enlargement. For single phase flows, the pressure drop across the restriction is a

straightforward function of the velocity and density of the fluid. For multiphase flows, the

analysis is more complicated. The gradual restriction in the flow path makes the Venturi

meter slightly intrusive to the flow.

Positive Displacement Meters (PD) – rely on the metered fluid to rotate mechanical gears or

rotors in the flow path. Each rotation of the rotor corresponds to a known amount of volume

passing through the meter. PD meters are commonly used in single phase service. For full

well stream production, risks due to erosion and blockage should be considered.


12

1.5 MFMs Projects

A very limited amount of information is available on MFMs performance. In the oil and gas

sector where competition is always intense a “black box” MFMs packages are usually offered,

where very little is unveiled. A brief description of some MFMs projects that are now

commercially available is given in Appendix A, together with the tables containing

comparison of the methods with regard to the techniques that are used for measurement

purposes.

Chapter 2

Pressure Pulse Technology

2. Pressure Pulse Technology __________________________________________________________________________

14

2.1 Pressure Pulse method

Multiphase metering in oilfield operation is of considerable interest in petroleum industry as

described in Chapter 1. As the response for these needs new method called Pressure-Pulse

has been developed at NTNU by Professor Gudmundsson. The method is based on the

propagation properties of pressure waves in gas-liquid media. Waves generated in gas-liquid

mixture flowing in a pipe at a speed of sound will propagate as pressure pulses

(Gudmundsson and Celius, 1999). These effects called water-hammer and line packing are

described precisely below. The method has been tested in several offshore platforms

including Gullfalks A, Gullfalks B, and Oseberg B, with positive repeatable results similar to

the theoretical models (Gudmundsson, Falk, 1999). Total of 800 tests were run on 12

different gravel packed wells. No negative effects were observed on the production system or

the reservoir during the 11-month test period. The method has the advantage of being simple,

low-cost, and gives the same accuracy as the competitors (Gudmundsson and Celius, 1999).

Pressure is the easiest parameter to measure in the production of oil and gas. It can be

measured in pipelines, flowlines and wellbores; at wellhead, chokes, manifolds, and

separators. The widespread use of the quick acting valves in the oil industry to open, close,

and control pipeline and wellbore flow, has made it possible to harness the information

contained in the rapid pressure transients when a valve is activated (Gudmundsson et al.,

2002).

2.2 Water-hammer effect

The water-hammer effect can be caused by a rapid closure a valve in pipe line with flowing

liquid. The immediate pressure increase created by the valve is referred as the acceleration

pressure-pulse ∆pa. Wylie and Streeter (1993) described how this increase in pressure travels

in the pipe with the velocity of sound, and stop the flow as it passes. The instant the valve is

closed, the fluid immediately adjacent to it is brought to rest by the impulse of the higher

pressure developed at the face of the valve. As soon as the first layer is stopped, the same

action is applied to the next layer of fluid bringing it to rest. In this manner a pulse wave of


15

high pressure is visualised as travelling upstream at same sonic velocity. However in long

pipe flows with high frictional pressure loss the accelerational pressure-transient is attenuated

and does not stop the flow completely. Yet, since the fluid must stop by the valve, there is a

continuous pressure increase near the valve also after is wholly closed. The name of these

phenomena is line packing. Figure 2.1 illustrates the water-hammer effect.

2.3 Theory and equation

Water-hammer phenomena, line packing and pressure pulse velocities are essential for the

new multiphase method. Water-hammer pressure transient can be found using homogenous

continuity equation at high pressure well conditions fluids are well mixed and thus

homogenous continuity equation can be applied (Falk, 1999).

Continuity equation

02 =∂∂

+∂∂

⋅+∂∂

xpu

xua

tp ρ (2.1)

The equation may be rewritten in form

02 =∂∂

∂∂⋅+

∂∂

∂∂

⋅+∂∂

xt

tpu

xt

tua

tp ρ (2.2)

The characteristic pressure pulse velocity running upstream the valve is uatx

−=∂∂ .

Thus,

02

=∂∂

−+

∂∂

−⋅

+∂∂

tp

uau

tu

uaa

tp ρ (2.3)

tua

tp

∂∂

⋅−=∂∂

ρ (2.4)


16

During quick valve closure the velocity jump is ∆u = -u in a short period of time ∆t. The

water-hammer due this retardation is

uapa ⋅⋅=∆ ρ (2.5)

This equation is generally known in literature as Joukowski equation.

Momentum conservation principle is given as

dxdzg

duu

fxp

xuu

tu

⋅−⋅

⋅⋅=

∂∂

+∂∂⋅+

∂∂

21ρ

(2.6)

In steady-state turbulent pipe flow frictional pressure gradient is represented by Darcy-

Weisbach equation

2

2u

df

Lp f ⋅⋅

⋅=

∆ρ (2.7)

where, f is the dimensionless friction factor.

The frictional pressure gradient is made available to measure when the flow is brought to the

rest after valve closure. The line-packing pressure increase, in liquid-only flow represents the

pressure drop with distance in the pipeline. In two-phase flow line-packing is more

complicated and in addition to frictional pressure gradient it contains also increase in water-

hammer with upstream distance. In vertical gas-liquid wells pressure increase with depth and

hence the water-hammer changes with depth (Gudmundsson and Celius, 1999).

2.4 Pressure surge in wellbores

Using a high sampling rate and high resolution pressure gauge, pressure buildup is possible to

record. A typical pressure-pulse technology set up is shown in Figure 2.2. It contains a quick-


17

acting valve and two pressure transducers A and B upstream of the valve taking samples in

micro to mili seconds time period. Today technology can definitely provide such high

sampling gauges. A valve is termed quick-acting if it closes completely before waves are

reflected from up-stream or downstream. If there are reflections before valve is closed, the

pressure on closing will be affected (Gudmundsson and Falk, 1999). The example of

measured pressure from two transducers is shown on Figure 2.3. Pressure Pulse is measured

at two locations spaced 83.35m up-stream a quick acting valve. The speed of sound may be

estimated from cross correlation between the signals. In this example figure this is 170 m/s.

By knowing the mixture density, acoustic velocity for the mixture and pressure increase due

to acoustic term during a quick shut-in, mixture velocity may be calculated at the wellhead

(Gudmundsson, 1999). The studies of Khokhar (1994) suggest that the phenomena like

wellbore storage, skin effect, and phase redistribution that occur after the well shut have no

effect on the pressure technique. Whereas, the pressure-pulse method dependents more on

mixture composition and gas-liquid ratio of the well fluids which influence the acoustic

velocity. The speed of sound in two-phase mixtures, and its dependency on fluid properties

and PVT conditions will be investigated in Chapter 5 of this work. In the Pressure-Pulse

method the sound speed can be determined from cross-correlation of two pressure signals

from locations A and B, as indicated in Fig. 2.4. The testing of the Pressure-Pulse method on

several North Sea fields has resulted in measurements that make this possible (Gudmundsson,

Falk, 1999).

2.5 Mass and volume flowrates

The mass flowrate in a pipe of constant cross-sectional area can be obtained directly from the

Joukowski water-hammer equation, when the sound speed is also determined from cross-

correlation of the measured delay time between two signals from transducers A and B.

⎥⎦⎤

⎢⎣⎡⋅∆=

skg

aApm a (2.8)


18

The continuity principle dictates that the mass flow rate at the valve is the same as the mass

flow rate at other locations. Mixture density and the mixture velocity can be also obtained

from the measurements

⎥⎦⎤

⎢⎣⎡

∆⋅⋅⋅∆⋅⋅

= 32

2

2 mkg

padpLf

f

amixρ (2.9)

⎥⎦⎤

⎢⎣⎡

∆⋅⋅

∆⋅⋅⋅= 3

2mkg

pLfpad

va

fmix (2.10)

Knowing the density of individual phases of the fluid mixture, void fraction can be calculated

gL

mixL

ρρρρ

α−−

= (2.11)

Flow rates in petroleum industry are traditionally expressed in volumetric quantities.

The mass flowrate and the volumetric flowrate of the liquid are related trough relationship

⎥⎦

⎤⎢⎣

⎡=

sSmmq

3

ρ (2.12)

Treating about volumetric flow rates requires volumetric factor to be taken into consideration.

Volumetric factor B(p,T), indicates the effects of pressure and temperature changes, from

reservoir to stock-tank conditions. Thus, volumetric flowrates for oil can be calculated as

( ) ⎥⎦

⎤⎢⎣

⎡⋅=

sSmTpBTpq ooo

3

,),( ρ (2.13)

where oil density given by relationship


19

( ) ⎥⎦⎤

⎢⎣⎡⋅+

= 3,),(

),(mkg

TpBTpR

Tpo

sgoo

ρρρ (2.14)

The volumetric flowrates of gas

( ) ⎥⎦

⎤⎢⎣

⎡⋅=

sSmTpBqTpq ogg

3

,),( (2.15)

where gas volume can be also expressed as

( ) ( )[ ] ( ) ⎥⎦

⎤⎢⎣

⎡⋅−=

sSmTpBTpRGORqTpq gsog

3

,,, (2.16)

where:

),( TpR - amount of dissolved gas in oil ⎥⎦

⎤⎢⎣

⎡oilSmgasSm

3

3

GOR – gas oil ratio at standard conditions ⎥⎦

⎤⎢⎣

⎡3

3

mSm

If the oil produced contains water, also watercut WC [%] need to be known.

2.6 Flow condition analysis

The pressure profile in a pipeline can be used to detect and monitor solid deposits as shown

on Figure 2.5. Deposits will change the frictional pressure drop in the affected interval both

by change pipe roughness and by reducing the tubing diameter. This will show up as increase

in the line packing gradient in the affected region. When the valve is activated the pressure is

measured resulting in a pressure time log. The pressure - time log is then converted into

pressure - distance log. Those give the location and extend of the deposits in a pipeline.

Pressure Pulse testing can be also used in gas lift wells for flow rate metering and flow

conditions analysis. An examination of the line packing pressures makes it possible to

identify the location of gas injection points, and asses the status gas lift valves (Gudmundsson


20

et al., 2002). Figure 2.6 shows an example of the simulations for three different valve

locations.

The bubble point depth may be identified from line packing as it appears with the peak on the

time derivative plot. Figure 2.7 shows typical bubble point pressure response experienced

during the fields tests.

2.7 Concluding remarks

The water-hammer theory treats pressure-pulse propagation in single phase flow, and has also

been directly extended to multiphase flow. This theory is important for the new multiphase

meter. Multiphase flow models like the drift flux model, the homogenous model and certain

forms of the two-fluid models could predict pressure pulse. However, the assumptions of the

one pressure in the one-dimensional two-fluid model are not appropriate. This thesis uses the

homogenous model, where due to large pressure surge fluid homogeneity and continuity may

be assumed. The multiphase models are described in Chapter 4 of this work that treats about

multiphase flow in wells.


21

Figure 2.1 Water Hammer Effect

Figure 2.2 Pressure Pulse setup


22

Figure 2.3 Pressure Pulse measurements (Gudmundsson and Celius, 1999)


23

Figure 2.4 Pressure Pulse technology principles


24

Figure 2.5 Deposit appearances on line packing


25

Figure 2.6 Simulation results for three different valve locations (Gudmundsson et al, 2001)


26

Figure 2.7 Typical bubble point pressure response during the field tests.

Chapter 3

Geothermal Applications

3. Geothermal Applications __________________________________________________________________________

28

3.1 Geothermal energy

Geothermal energy is one of the cleaner forms of energy now available in commercial

quantities. The use of this alternative energy source, with low atmospheric emissions, has a

beneficial effect on our environment by displacing more polluting fossil and nuclear fuels.

Thermal energy carried in the produced fluid can be used for direct heating in residential,

agricultural, and industrial applications; or the thermal energy of higher temperature systems

can be used to produce electricity. Rapidly growing energy needs around the world will make

geothermal energy exceedingly important in several countries. For example in Iceland

provides 50% of the total power supply, and 86% energy used for space heating (Ragnarsson,

2000). The production of electricity requires a greater concentration of energy than other

applications. If hot fluid is available in great enough quantities, a geothermal power plant can

be installed that uses the produced steam directly to drive a turbine generator system.

Geopressured geothermal reservoirs are closely analogous to the geopressure oil and gas

reservoirs. Fluid caught in stratigraphic trap may be raised to litostratic pressure due to

overburden pressure. Such reservoirs are given fairly deep (over 2,000 m), so that the

geothermal gradient can give temperature over 100oC (Grand, 1982). A number of such

reservoirs have been found in drilling for oil and gas. These reservoirs derive their heat from

the terrestrial heat flux, and are widespread throughout the world. It occurred not economic

to exploit even the most favourable reservoirs for a long time, but over the last decade many

projects arise to utilise their energy (Dickson and Fanelli, 2001).

In some places over the world high temperature over 250 oC geothermal reservoirs occur.

That heat source may be either an abnormal high geothermal gradient or volcanism nature.

Those fields usually display surface activity when high temperature fluid systems transfer

heat to the surface from crustal rocks heated by magmas and are mainly located in six

countries: United States, Mexico, New Zealand, Philippines, Iceland and Italy

(Gudmundsson and Ambasth, 1986).


29

3.2 Geothermal well flow

High temperature geothermal reservoirs can be liquid and vapour dominated this mean that

can have liquid only or steam-water feedzone (Gudmundsson, 1989). Steam dominated

reservoirs are relatively rare, and most geothermal fields are water-dominated, where liquid

water at high temperature, but also under high (hydrostatic) pressure, is the pressure-

controlling medium filling the fractured and porous rocks. When liquid water flows into a

geothermal well, the water will remain liquid up the wellbore until reaching a depth where

the pressure is equal to the saturation pressure. The pressure decreases as the water moves

toward the surface and at this depth the liquid water will start to flash to form a steam. It will

continue to flash until reaching the wellhead, surface pipeline, and eventually the steam

separator. Beginning with liquid water, the first flashing results in comparatively small

amounts of stream that flows as a bubbles trough a continuous column of water. With

pressure drop towards the surface more steam evaporates and thus flow changes the regime

into slug and steam continuous annular flow (Gudmundsson, 1989).

The two-phase output from geothermal wells is piped to a separator to produce steam for

electric power generation. The liquid water separated from the steam is disposed of at the

surface or injected back into the reservoir. Reinjection of the geothermal liquid back into

reservoir after use has a number of purposes. The most important ones being (Eliasson, 2001):

- disposal being use liquid without polluting the environment,

- sustenance of the reservoir pressure to counteract with drawn down and

surface subsidence,

- mining of heat stored in hot formations simultaneously extend the useful life

of the reservoir.

The most common approach for measuring flow rate similarly to the petroleum industry

where gas is separated from oil, in geothermal applications separator is also used, where

steam-water mixture is separated into a flow of water and steam at the pressure of separator.

The flow of each phase can be then measured individually using pressure differential devices.


30

High temperature wells are typically drilled in four stages: (Figure 3.1)

- a wide hole to a depth of 50-100 [m] into which is cemented the surface casing.

- a narrower hole to a depth of 200-600 [m] into which the anchor casing is cemented.

- a narrower hole still to a depth of 600 to 1,200 [m] which carries a cemented casing

called the production casing

- finally the production part of the wells drilled into and/or trough the active aquifer.

This part carries a perforated liner that is hung from the production casing reaching

almost to the well bottom.

On top (wellhead) the well is fitted with expansion provision and a sturdy sliding plate valve

(master valve). It is also commingled to a muffler, usually of a steel cylinder fitted with an

expanding steam inlet pipe to low down the fluid on entry. Steam capacity of these wells

commonly range between 3 ÷ 30 [kg/s] (1.5 MWe – 15 MWe) (Eliasson, 2001).

The drilling programs are of two types:

Standard:

- Surface casing 18’’ nominal diameter in a 20’’ hole.

- Anchor casing 13 3/8’’ nominal diameter in a 17 1/2’’ hole.

- Production casing 9 5/8’’ nominal diameter in a 12 1/2’’ hole

- Liner 7’’ nominal diameter in an 8 1/2’’ hole.

Wide:

- Surface casing 22’’ nominal diameter in a 24’’ hole.

- Anchor casing 18’’ nominal diameter in a 20’’ hole.

- Production casing 13 3/8’’ nominal diameter in a 17 1/2’’ hole

- Liner 9 5/8’’ nominal diameter in a 12 1/2’’ hole.

Wide tubing configurations has been initially implemented in order to cut down the

frequency of wellbore cleaning due to calcium carbonate scale depositions (Gudmundsson,

1986). These wide 13 3/8” production casing has been particularly developed in Iceland and


31

the narrower 9 5/8’’ production casing is reported by many authors as the typical (Uphady et

al., 1977), (Gudmundsson and Thrainsson, 1988) . In present work, data about the wells was

taken from Icelandic sources for 13 3/8” production casing, from two fields Reykjanes and

Svartsengi. Nevertheless the deliverability considerations, presented in next section allowed

finding the operating parameters assuming the 9 5/8’’ production casing and thus simulations

covered the both typical tubing sizes.

Geothermal wells have total mass flowrates are greater than oil and gas wells, primarily due

to width casing configuration presented above that allows yield such high mass flowrates.

The calculations made in this work confirmed that diameter change from 13 3/8” to 9 5/8’’

allows yield almost double output.

Typical exploitation parameters gained from literature are placed in the Table 3.1

Table 3.1 Typical exploitation parameters

Variables Range

total mass flowrate 12.9 - 68.6 [kg/s]

wellhead pressure 2.3 - 56.5 [bar]

wellhead temperature 150 - 250 [oC]

wellhead enthalpy 965 - 1966 [kJ/kg]

well depth 913 - 2600 [m]

Geothermal reservoir data inspection presented here is required to characterise typical

geothermal well that can be used in pressure pulse simulations. Simulations will be done for

various mass flowrates, and wellhead pressures. In order to predict the above parameters,

deliverability method developed in petroleum industry and widely applied also for

geothermal reservoir engineering is necessary.


32

3.3 Well performance

The production of liquid water from a geothermal reservoir depends on the reservoir pressure,

the flow of fluid trough the feedzone into the well, and then up the wellbore to the surface.

These three elements of deliverability are called reservoir, inflow and vertical lift

performance respectively. The production output test gives the deliverability at the time of

testing. As the production proceeds with the time the deliverability is like to change because

of drawdown in reservoir pressure. The prediction of the reservoir pressure with time is the

subject of reservoir modelling, and is not necessary to be discussed here.

The fluid entering flowing well in liquid dominated reservoir contains pressurized water.

Nevertheless when well flowing pressure pwf decrease below saturation pressure psat a two-

phase mixture of steam vapor and liquid water flows into the wellbore as a result of flashing

outside the wellbore. The flashing occurs over a relatively short distance near the wellbore.

This indicates that rapid pressure drop and radial-flow effects in the wellbore region may

control the output characteristics of geothermal well (Gudmundsson, 1986). The inflow

performance curve for geothermal well is composed of two forms of flow behavior,

depending upon whether the flowing pressure is above or below the saturation pressure of the

geothermal fluid. Above the saturation pressure a linear relationship as assumed between the

mass flowrate m and the well flow pressure pwf.

The inflow performance curve for geothermal well is composed of two forms of flow

behavior, depending upon whether the flowing pressure is above or below the saturation

pressure of the geothermal fluid. Above the saturation pressure a linear relationship is

assumed between the mass flowrate m and the well flow pressure pwf. In general the mass

flowrate increase when pressure difference enlarges as can be expressed:

( ) ⎥⎦⎤

⎢⎣⎡−⋅=

skgppPIm wfr (3.1)

where:


33

m - mass flowrate ⎥⎦⎤

⎢⎣⎡

skg

rp - average reservoir pressure [ ]bar

wfp - well flow pressure [ ]bar

PI - productivity index ⎥⎦⎤

⎢⎣⎡

⋅ sbarkg

This equation applies for single-phase Darcy flow into the wellbore.

If the pressure in a near wellbore distance decreases below bubble point pressure the slope of

inflow performance curve is assumed to become more negative. This indicates that when

steam-water mixture enters the wellbore, the resistance to flow is grater than for liquid only

flow for the same flowrate. It is like a solution-gas drive reservoir in petroleum industry, and

thus equation from petroleum industry can be adopted, after some modifications. The orginal

form of the equation for oil and gas is given below

( ) ( )⎥⎦

⎤⎢⎣

⎡⋅

−⋅

+−⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅⋅⋅⋅

+−⋅+−⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅⋅⋅⋅

=s

smppp

srr

Bk

hk

pps

rr

Bhk

qb

wfb

w

e

poo

or

br

w

e

pooo

br322

243ln

2

43ln

12µ

πµ

π

(3.2)

where:

k – permeability [m2]

h – thickness [m]

µ – dynamic viscosity [Pa·s]

re – effective radious [m]

rw – well radious [m]

Using HOLA 3.3 simulator described in Chapter 6, it is possible to estimate Productivity

Index PI for given wellhead flow conditions.


34

In Darcy law for two phase flow the fluids are assumed to flow practically independently of

each other. The fundamental law is then applied to the two phase flow individually. In

geothermal simulation studies of two phase reservoir flow, the relative permeability for

steam and water need to be defined. The following expression gives the total mass flowrates:

⎥⎦⎤

⎢⎣⎡⋅

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⋅⋅−=s

kgdLdpkk

kAm

satps

s

sr

w

w

wr

ρµ

ρµ

(3.3)

Thus, to calculate curve for liquid only feedzone, when well flow pressure pwf above

saturation pressure, the following equation can be used:

( ) ⎥⎦⎤

⎢⎣⎡−⋅=

skgppPIm wfr (3.4)

And as the well flow pressure pwf decrease below saturation pressure the equation (3.3) can

be used in form

( ) ( )⎥⎦

⎤⎢⎣

⎡⋅

−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⋅+−⋅=s

smp

ppkkPIppPIm

sat

wfsat

satps

s

sr

w

w

wrsatr

322

2ρµ

ρµ

(3.5)

where: psat – saturation pressure [bar]

Gudmundsson et al. (1986) in their study of relative permeabilities give the necessary

relations. The relative permeability ratio of vapor and water can be calculated from equation

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟

⎠⎞

⎜⎝⎛=

w

w

s

w

sr

wr

SS

Kkk

11

µµ

(3.6)


35

where: K - slip ratio,

Sw – water saturation;

Assuming that there is no interaction between the flowing phases, that is steam and water are

assumed to flow independently, the retaliations can be made

1=+ srwr kk (3.7)

Relative permeability for steam and water can be found from following functions:

For ,4.0<wS 6,0wwr Sk = ;

for ,4.02.0 << wS 7,0wwr Sk = ;

and

for ,2.0<wS 77,0wwr Sk = ;

The other way to calculate the two-phase performance curve part, below the saturation

pressure is to use the Vogel empirical relationship obtained for the situation when gas is

coming out of the solution (Gudmundsson, 1986).The equation has form

2

max

8.02.01 ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅−=

sat

wf

sat

wf

pp

pp

mm (3.8)

Where mmax is the ideally maximum flowrate obtained assuming pwf = 1[bar]

Vertical lift performance curves were calculated for particular wellhead pressures pwh, using

HOLA 3.1 wellbore simulator. Then MATHLAB 6.5 program has been used to calculate the

deliverability curves. The mass flowrate of steam and water from geothermal reservoir-

wellbore system is given by well operating point, determined by the intersection of the IPR

and VLP curves.


36

Figure 3.1 Casing stage types

Chapter 4

Multiphase Flow in Wells

4. Multiphase Flow in Wells __________________________________________________________________________

38

4.1 Introduction

Two-phase flow occurs commonly in the petroleum, geothermal, chemical, civil, and nuclear

power industries. In the petroleum and geothermal industry, two-phase flow is encountered in

well production, transportation, processing systems. The complex nature of two-phase flow

challenges production engineers with problems of understanding, analyzing, and modelling

two-phase-flow systems. The calculation and prediction methods that are discussed in this

chapter were developed for petroleum industry. Geothermal applications also base on this

method, however due to different water and crude nature a different approach is required in

some cases.

4.2. Main difficulties

When two or more phases flow simultaneously in pipes, the flow behaviour is much more

complex than for single-phase flow. Phases tend to separate because of differences in density.

Shear stresses at the pipe wall are different for each phase as a result of their different

densities and viscosities. Expansion of the highly compressible gas phase with decreasing

pressure increases the in-situ volumetric flow rate of the gas. As a result, the gas and the

liquid phases normally do not travel at the same velocity in the pipe, upward flow the less

dense, more compressible, less viscous phase tends to flow at a higher velocity than the liquid

phase, causing a phenomenon known as slippage. However, for down flow, the liquid often

flows faster than the gas.

Perhaps the most distinguishing aspect of multiphase flow is variation in the physical

distribution of the phases in the flow conduit characteristic known as flow pattern or flow

regime (Brill, 1999). During multi-phase flow through pipes, the flow pattern that exists

depends on the relative magnitudes of the forces that act on the fluids. Buoyancy turbulence,

inertia, and surface-tension forces vary significantly with flow rates, pipe diameter,

inclination angle, and fluid properties of the phases (Brill, 2001). Several different flow

patterns can exist in a given well result of the large pressure and temperature changes the

fluids encounter (Manabe et al., 2001). Especially important is the significant variation in


39

pressure gradient with flow pattern. Thus, the ability to predict flow pattern as a function of

the flow parameters is of primary concern.

Analytical solutions are available for many single-phase flow problems. Even when empirical

correlations were necessary (i.e., for turbulent-flow friction factors), the accuracy of

prediction was excellent. The increased complexity of multiphase flow logically resulted in a

higher degree of empiricism for predicting flow behaviour. Many empirical correlations have

been developed to predict flow pattern, slippage between phases, friction factors, and other

such parameters for multi-phase flow in pipes. Virtually all the existing standard design

method relies on these empirical correlations. However, since the mid-1970’s, a dramatic

advance have taken places that improve understand the fundamental mechanisms that govern

multiphase flow. These have resulted in new predictive methods that rely much less on

empirical correlations.

This chapter introduces and discusses basic definitions for parameters unique to multiphase

flow in pipes. Flow patterns are described in detail, including methods available to predict

their occurrence. The use of empirical correlations based on dimensional analysis and

dynamic similarity performed by software used in this work are presented.

4.3. Phase behaviour

Two-phase can be interpreted as a single component like a water and its vapour – steam, and

a complex mixture of various components like a hydrocarbons composition. Geothermal fluid

or complex mixture of hydrocarbon compounds or components can exists as a single-phase

liquid, a single-phase gas, or as a two-phase mixture, depending on the pressure, temperature,

and the composition of the mixture (Campbell, 1994).

Unlike to a single component or compound, such as water-steam system, when two phases

exist simultaneously a multicomponent mixture will exhibit an envelope rather than single

line on a pressure/temperature diagram. Figure (3.1) gives a typical phase diagram for a

multicomponent hydrocarbon system. Shapes and ranges of pressure and temperature for


40

actual envelopes vary widely with composition. Figure (3.1) permits a qualitative

classification of the types of reservoirs encountered in oil and gas systems.

Typical oil reservoir has temperatures below the critical temperature of the hydrocarbon

mixture. Volatile oil and condensate reservoirs normally have temperatures between the

critical temperature and the cricondentherm for the hydrocarbon mixture. Dry gas reservoirs

have temperature above the cricondentherm (Campbell, 1994). Many condensate fluids

exhibit retrograde condensation, a phenomena in which condensation occurs during pressure

reduction rather than with pressure increase, as for most gases (Firoozabadi, 1999). This

abnormal or retrograde behaviour occurs in a region between the critical and the

cricondentherm, bounded by the dewpond curve above and, a curve below formed by

connecting the maximum temperate for each liquid volume percent.

As pressures and temperatures change, mass transfer occurs continuously between the gas

and the liquid phases within the phase envelope of Fig. 3.1. All attempts to describe mass

transfer assume that equilibrium exists between the phases. Two approaches have been used

to simulate mass transfer for hydrocarbons the "black-oil" or constant-composition model and

the (variable) compositional model (Brill, 1999). Each is described in the following sections.

Figure 3.1 Typical phase diagram (Campbell, 1994)


41

Black-Oil Model

The term black oil is a misnomer and refers to any liquid phase that contains dissolved gas,

such as hydrocarbons produced from oil reservoirs. These oils are typically dark in colour,

have gravities less than 40° API (824.97 kg/m3), and undergo relatively small changes in

composition within the two-phase envelope (William and McCain, 2002). A better

description of the fluid system is a constant-compositional mode. For black oils with

associated gas, a simplified parameter Rs has been defined to account for gas that dissolves

(condenses) or evolves (boils) from solution in the oil. This parameter, Rs can be measured in

the laboratory or determined from empirical correlations. Because the black-oil model cannot

predict retrograde condensation phenomena, it should not be used for temperatures

approaching the critical-point temperature.

A second parameter, called the oil formation volume factor Bo also has been defined to

describe the shrinkage or expansion of the oil phase. Oil volume changes occur as a result of

changes in dissolved gas and because of the compressibility and thermal expansion of the oil.

Dissolved gas is by far the most important factor that causes volume change. Oil formation

volume factor can be measured in the laboratory or predicted with empirical correlations

(Brill and Mukherjee, 1999). Once the black-oil-model parameters are known, oil density and

other physical properties of the two phases can be calculated. When water also is present,

solution gas/water ratio, Rsw, and water formation volume factor, Bw, can be defined. Brill and

Mukherjee (1999) also give correlations for these parameters and physical properties of the

water. The amount of gas that can be dissolved in water and the corresponding possible

changes in water volume are much smaller than for gas/oil systems (William and McCain,

2002).

Compositional Model

For volatile oils and condensate fluids, vapour-liquid equilibrium (VLE) or "flash"

calculations are more accurate to describe mass transfer than black-oil-model parameters.

Brill and Mukherjee (1999) provide a description of VLE calculations. Given the

composition of a fluid mixture or "feed," a VLE calculation will determine the amount of the

feed that exists in the vapour and liquid phases and the composition of each phase. From


42

these results, it is possible to determine the quality or mass fraction of gas in the mixture.

Once the composition of each phase is known, it also is possible to calculate the interfacial

tension and densities, enthalpies, and viscosities of each phase. Brill and Mukherjee (1999)

also give methods to predict these properties.

VLE calculations are considered more rigorous than black-oil model parameters to describe

mass transfer. However, they also are much more difficult to perform. If a detailed

composition is available for a gas/oil system, it is possible to generate black-oil parameters

from VLE calculations. However, the nearly constant compositions that result for the liquid

phase and the increased computation requirements make the black-oil model more attractive

for non-volatile oils (Brill, 1999).

4.4. Definition and variables

When performing multiphase calculations, single-phase flow equations often are modified to

account for the presence of a second phase. This involves defining mixture expressions for

velocities and fluid properties that use weighting factors based on either volume or mass

fraction (King, 1990).

When gas and liquid flow simultaneously up a well, the higher mobility of the gas phase

tends to make the gas travel faster than the liquid. This is a result of the lower density and

viscosity of the gas. The slippage between both phases in defined as the ratio of the gas

velocity to the liquid velocity

L

G

uu

K = (4.1)

where, uG – gas velocity [m/s], uL – liquid velocity [m/s].

The mass fraction of flowing phases is defined as the ratio of gas mass flowrate to the total

mixture flowrate:


43

LG

G

mmm

x+

= (4.2)

where, mG – gas mass flowrate [kg/s], mG – liquid flowrate [kg/s]. The gas mass flowrate is

related to the volume flowrates with expression

GGGG Aum ⋅⋅= ρ (4.3)

and similarly the liquid phase

LLLL Aum ⋅⋅= ρ (4.4)

where AG and AL are the cross sectional area occupied by gas and liquid phase respectively.

Under steady state condition the slippage between both phases result in a disproportionate

amount of the slower phase being present at any given location in the well. Gas void fraction

can be defined as the fraction of pipe cross sectional area occupied by gas. Substitution of the

equations (4.3) and (4.4) to the equation (4.2) results in the void fraction given by

( )xKx

x

L

G −⋅⋅+=

1ρρ

α (4.5)

The opposite value to the gas void fraction is the liquid holdup defined similar way as the

cross section area occupied by liquid or volume increment that is occupied by the liquid

phase

( )α−= 1LH (4.6)

The gas void fraction and liquid holdup can be distinguished in horizontally oriented pipes

where stratification occurs due to gravity. In vertical wellbore two-phase turbulent flow under


44

high velocities, both phases may be considered as a homogenous mixture (King, 1990). Two

phases may be assumed to flow at the same mixture velocity with no slippage between.

4.5. Fluid properties

A numerous equations have been proposed to describe the physical properties of gas/liquid

mixtures. The following expression has been used to calculate in multi-phase flow mixture

density

( ) LGM ραραρ ⋅−+⋅= 1 (4.7)

The two phase viscosity is the property expressed per mass unit and thus was calculated from

equation

LGM

xxµµµ−

+=11 (4.8)

When performing the temperature change calculations for multi-phase flow in geothermal

wells, it is necessary to predict the enthalpy of the multiphase mixture. Also most VLE

calculation method for oil wells includes a provision to predict the enthalpies of the gas and

liquid phases. Enthalpy of the mixture was calculated from equation

( ) LGt hxhxh ⋅−+⋅= 1 (4.9)

4.6. Flow patterns

Prediction the flow pattern that occurs at a given location a well is extremely important. The

empirical correlations or mechanic model used to predict flow behaviour varies with flow

pattern (Gomez, 2001). Essentially all flow pattern predictions are based on data from low-

pressure systems, with negligible mass transfer between the phases and with a single liquid

phase (Brill, 1999). Consequently, these predictions may be inadequate for high-pressure,


45

high production-rates, evidently high-temperature geothermal wells, or for wells producing

oil and water or crude oils with foaming tendencies, respectively (Manabe et al., 2001),

(Gudmundsson and Ambastha, 1984), (Aggour, 1996).

A consensus exists on how to classify flow patterns (Brill, 1999). For upward multi-phase

flow of gas and liquid, most investigators now recognize the existence of four flow patterns:

bubble flow, slug flow, churn flow, and annular flow. These flow patterns, shown

schematically in Fig. (4.2) and Figure (4.3) are described next. Slug and churn flow are

sometimes combined into a flow pattern called intermittent flow. It is common to introduce a

transition between slug flow and annular flow that incorporates churn flow. Some

investigators have named annular flow as mist or annular-mist flow.

Flow in vertical and horizontal or inclined pipes exhibits different behaviour. The distribution

of the multiphase contents across the pipe in vertical flow regimes is randomly chaotic, and

the phases show no preferences for the one side of the pipe or another. The exception to the

random distribution is annular flow where at very high flow rates gas occupies the centre of

the pipe. There may be large discontinuities that pass along the vertical pipe or wellbore, as

when gas flows much faster than liquid in slug and churn flow regime. In non vertical flow

random distribution of the phases across the pipe is replaced by gravity segregation by the

phases.

Bubble Flow Bubble flow is characterized by a uniformly distributed gas phase and discrete

bubbles in a continuous liquid phase. Based on the presence or absence of slippage between

the two phases, bubble flow is further classified into bubbly and dispersed - bubble flows. In

bubbly flow, relatively fewer and larger bubbles move faster than the liquid phase because of

slippage. In dispersed bubble flow, numerous tiny bubbles are transported by the liquid phase,

causing no relative motion between the two phases.

Slug Flow Slug flow is characterized by a series of slug units. Each unit is composed of a

gas pocket a plug of liquid called a slug, and a film of liquid around the bubble flowing

downward relative to the Taylor bubble. The Taylor bubble is an axially symmetrical, bullet-


46

shaped gas pocket that occupies almost the entire cross-sectional area of the pipe. The liquid

slug, carrying distributed gas bubbles, bridges the pipe and separates two consecutive Taylor

bubbles.

Churn Flow Churn flow is a chaotic flow of gas and liquid in e which the shape of both the

Taylor bubbles and the liquid slugs are distorted. Neither phase appears to be continuous. The

continuity of the liquid in the slug is repeatedly destroyed by a high local gas concentration.

An oscillatory or alternating direction of motion in the liquid phase is typical of churn flow.

Annular Flow Annular flow is characterized by the axial continuity of the gas phase in a

central core with the liquid flowing upward, both as a thin film along the pipe wall and as

dispersed droplets in the core. At high gas flow rates more liquid becomes dispersed in the

core, leaving a very thin liquid film flowing along the wall.

The interfacial shear stress acting at the core/film interface and the amount of entrained liquid

in the core are important parameters in annular flow.

Figure 4.2 Vertical flow patterns


47

Figure (4.3) Horizontal and inclined flow patterns

4.7. Pressure gradient

The pressure gradient equation for multi-phase flow can be modified from single-phase flow.

Considering the fluids to be a homogenous mixture the equation may be written

dLdu

ugd

ufdLdp M

MMMMM ⋅⋅+⋅⋅+

⋅⋅⋅

= ρθρρ

sin2

2

(4.10)

For vertical flow θ = 90o, dL =dz and the equation for pressure gradient can be written as

accelf dzdp

dzdp

dzdp

dzdp

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ (4.11)


48

The pressure-gradient equation for single-phase flow in pipes was developed by use of the

principles of conservation of mass and linear momentum. The same principles are used to

calculate pressure gradient for multiphase flow in pipes. However, the presence of an

additional phase makes the development much more complicated. The pressure-drop

component caused by friction loses requires evaluation of two phase friction factor. The

pressure drop caused by elevation change depends on the density of the two phase mixture

which may be calculated from equation (4.7). The pressure drop caused by acceleration

component in normally negligible as is considered only for cases of very high flow velocities

(King 1990).

4.8. Multiphase flow models

Early investigators treated multiphase flow as a homogeneous mixture of gas and liquid. This

approach did not recognize that gas normally flows faster than liquid. The no slip approach

tended to under predict pressure drop because the volume of liquid predicted to exist in the

well was too small (Brill, 1999) Improvements to the no-slip methods used empirical liquid

holdup correlations to account for slippage between the phases. Although liquid holdup and

friction effects were often dependent on the flow pattern predicted by empirical flow-pattern

maps, in general these methods still treated the fluids as homogeneous mixture (Falk 1999).

Treating the fluids as a homogeneous mixture is often unrealistic, resulting in poor

predictions of flow behaviour (Brill, 1999). A trend to improve flow-behaviour predictions

has emerged that is a compromise between the empirical correlations and the two-fluid

approach. The methods used to predict pressure gradient can be classified as empirical

correlations and mechanistic models (Gomez, 1999). The empirical ones are based on

experimental data, and are suitable for preceding steady-state flow. The mechanistic

multiphase flow models include the two-fluid model, the drift-flux model and the

homogenous model (Manabe et al., 2001). These can be developed from physical relationship

with mass, momentum and energy conservation of each phase resulting in local,

instantaneous equations. The mechanistic modelling approach still requires use of some


49

empiricism, but only to predict specific flow mechanisms or closure relationships (Gomez,

1999).

The conservation laws are connected by interaction laws between the phases and between the

fluid and wall. A popular approach is to average the conservation equations over the cross

sectional area to get a one-dimensional model. The mechanistic models differ from each in

how they implement the conservation laws (Falk, 1999). The two-fluid method uses one

conservation equation for each phase, the drift-flux method uses the sum of the momentum

equations in addition to energy and mass conservation for each phase, while the homogenous

flow model uses only the sum of all phases for each conservation law. The homogenous flow

model is a simplification, assuming the same flow velocity for all phases (Gould, 1970), thus

needs neither interfacial friction nor drift flux terms.

The empirical correlations can be placed in one of three categories (Brill,1999):

I category - no slip, no flow pattern consideration. The mixture density is calculated based on

the input gas/liquid ratio. That is, the gas and liquid are assumed to travel at the same velocity.

The only a correlation required is for the two-phase friction factor. No distinction is made for

different flow patterns.

II category - slip considered, no flow pattern considered. A correlation is required for both

liquid holdup and friction factor. Because the liquid and gas can travel at different velocities,

a method must be improved to predict the portion of the pipe occupied by liquid at any

location. The same correlations used for liquid holdup and friction factor are used for all flow

patterns.

III category - slip considered, flow pattern considered. Not only are correlations required to

predict liquid holdup and friction factor, but methods to predict which flow pattern exists are

necessary. Once the flow pattern is established, the appropriate holdup and friction factor

correlations are determined. The method used to calculate the acceleration pressure gradient

also depends on flow pattern.


50

The following list presented on Figure (4.3) gives the published empirical correlations for

vertical upward flow and the categories in which they belong.

Method

Category

Poettmann and Carpenter

Baxendell and Thomas

Fancher and Brown

Hagedorn and Brown

Gray

Asheim

Duns and Ros

Orkiszewski

Aziz

Chierici

Beggs and Brill

Mukherjee and Brill

I I I

II

II

II

III

III

III

III

III

III

Figure 4.3 Published Vertical Flow Correlations Categories

The following sections of this chapter present method to predict pressure gradients and

presents the methods applied for calculations that were performed. In present work Duns and

Ros, 1963 for oil wells and also Orkiszewski, 1967 for geothermal well methods was applied.

Those authors summarized numerous investigations that have described flow patterns in

wells and made attempts to predict when occur. Both correlations were verified many times


51

since the time that was developed by other authors and by the industry elaborating the

software commercially available. Those methods are recommended for vertical wells

calculations and contain modifications that improve accuracy. (PipeSim Manual),

(Gudmundsson and Oritz, 1984) (Uphady, 1977)

4.9. Duns and Ros correlation for multiphase flow in oil wells

Duns and Ros method was chosen for oil wells calculations performed in present work. This

method is ranged to III group from Figure (4.3), which is assumed to give the most

appropriate issues. The method is a result of an extensive laboratory study in which liquid

holdup and pressure gradients were measured. About 4,000 two-phase-flow tests were

conducted in a 185-ft (56,39m)-high vertical-flow loop. Pipe diameters ranged from 1.26 to

5.60 in. and included two annulus configurations. Most of the tests were at near-atmospheric

conditions with air for the gas phase and liquid hydrocarbons or water as the liquid phase.

Liquid holdup was measured by use of a radioactive-tracer technique. A transparent section

permitted the observation of flow pattern. For each of three flow patterns observed,

correlations were developed for friction factor and slip velocity, from which liquid holdup

can be calculated. Duns and Ros performed the first dimensional analysis of two-phase flow

in pipes. They identified 12 variables that were potentially important in the prediction of

pressure gradient. Performing a dimensional analysis of these variables resulted in nine

dimensionless groups. Through a process of elimination, four of the groups were identified as

being important and were used to select the range of variables in the experimental program.

Equations presented in Appendix B for this method gives those four groups.

4.10. Duns and Ros modifications

Two proprietary modifications of the Duns and Ros method have been developed but are not

available in the literature. The first, known as the Ros field method, involved modifications

based on carefully obtained data from 17 high-GOR vertical oil wells. In a joint Mobil-Shell

study undertaken between 1974 and 1976, a modification resulted in the Moreland-Mobil-

Shell method (MMSM) (PipeSim Manual). In this study, 40 vertical oil wells, including the


52

17 used in the Ros field method, and 21 directional wells were selected as the basis for the

modifications. The MMSM method includes liquid- holdup correlations derived from the

data for bubble and slug flow that are simpler in form than those used in the original Duns

and Ros method (Brill, 1999). Possible discontinuities at flow-pattern-transition boundaries

also were removed.

4.11. Orkiszewski correlation for multiphase flow in geothermal wells

For Geothermal Two-phase flow calculations Orkiszewski correlation was used. This method

was recommended by Uphadhay and Hartz (1977). Their work contains comparison of

calculated and observed flowing pressure profiles for geothermal wells located in the United

States and Philippines. Comparisons were included for tubular flow as well as flow trough

the casing-tubing annulus. Their work revealed that for tubular flow, the Orkiszewski

correlation makes the best prediction, whereas for annular flow, no clear choice of correlation

can be made. Similar work has been done by Ambastha and Gudmundsson (1986), they

measured the flowing pressure profile data from many geothermal wells around the world,

covering a vide range of flowrate, fluid enthalpy and wellhead pressures. The authors

reported a good accuracy of Orkiszewski correlation in estimating the downhole conditions.

The capability to accurately predict flowing pressures in geothermal wells producing steam

and water a mixture under various operating conditions is of value importance for several

reasons similar to petroleum industry. These are general evaluation of the geothermal

reservoir under proper reservoir management, optimalisation of the wellbore design from

well deliverability considerations and minimization of scale deposits in the wellbore

(Ragnarsson, 2000). The predictive capability is especially important because of the difficulty

of running flowing pressure surveys in geothermal wells. These wells are characterized by

very high fluid velocities, which sometimes makes impractical for pressure recorders to

traverse downward in the well. There have been the cases of pressure recorders thrown out of

the wellbore due to fluid velocities (Uphadhay, 1977).

In calculating flowing pressure profiles in oil wells, phase transfer between oil and gas


53

requires a rather simple treatment, and is accomplished trough the use of solution gas-oil ratio

– Rs relationship. In geothermal wells, however phase transfer between water and steam

attains critical importance and calculations must incorporate the steam tables accurately.

Pressure profile calculations for geothermal wells vary from those for oil well in another

important aspect in that the temperature of the fluid must be computed precisely.

Orkiszewski tested several published correlations with field data and concluded that none

was sufficiently accurate for all flow patterns (Orkiszewski, 1967) He then selected what he

considered to be most accurate correlations for bubble and mist flow and proposed a new

correlation for slug flow. Orkiszewski used the Duns and Ros flow-pattern transition for the

boundaries between slug flow and mist flow, including the transition region between them.

Equation defined these are given in Appendix B together with Duns and Ros correlation

description. For the boundary between bubbly flows, he chose these criteria established by

Griffith and Wallis.

Chapter 5

Speed of Sound in Two-Phase Mixtures

5. Speed of Sound in Two-Phase Mixtures __________________________________________________________________________

55

5.1. Introduction

A book on physical acoustic (Trusler, 1991), defined sound as infinitesimal pressure waves

propagating trough a medium with a characteristic speed; the velocity of sound depending on

media. In present work a pressure wave caused by a rapid valve closure travels at the

velocity of sound trough the two phase mixture. This velocity is dependant on the

compressibility and densities of both phases. These flowing phases may have different

structures; usually it is of the gas bubbles or slug surrounded by the liquid phase. The gas

present in liquid phase cause a marked increase of damping. The examination of this

phenomena showed that this damping is due to increase of distortion of the liquid separating

the bubbles. The pressure variations act almost entirely on the volume of gas and scarcely at

relative incompressible liquid (Firoozabadi, 1999)

5.2. Compressibility of two-phase mixtures

The velocity of sound is defined as the square root of the derivative of pressure with respect

to the density at constant entropy (Henry et al. 1977)

S

pa ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=ρ

(5.1)

The second law of the thermodynamics tells that process must be isentropic due to there is no

temperature gradient except inside the wave itself. Therefore instead of differentiate the

density, the sonic velocity can be related to the properties of the fluid. Using isentropic

compressibility Ks the speed of sound formula can be written

SKa

⋅=ρ

12 (5.2)


56

The isentropic compressibility of a single fluid Ks is defined as

nSS p

vv

K,

1⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⎟⎠⎞

⎜⎝⎛−= (5.3)

where v is the specific volume [m3/kg] and n is the composition vector, which is defined by

n = (n1, n2, n3, …, nc), where c is the total number of components and ni is the number of moles

of each component i of the mixture. The expression for the two-phase gas liquid mixture

compressibility will be

nS

M

MMS p

vv

K,

1⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−= (5.4)

vM is the total specific volume of the gas liquid phases in the mixture. Similarly, the

isothermal compressibility of a two-phase multicomponent system is defined by

nT

M

MMT p

vv

K,

1⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−= (5.5)

The isothermal compressibility represents the volume change caused by small change in the

pressure of the closed system at constant temperature. The isothermal and isentropic

compressibility in the single phase state are related by a simple expression (Firoozabadi,

1999)

SV

pT K

CC

K ⋅= (5.6)

Where Cp and CV are the heat capacities [kJ/kg·K] at constant pressure and constant volume,

respectively. The derivation of (5.6) can be found in (Firoozabadi, 1999).


57

Since Cp ≥ Cv, then KT ≥ KS. In reservoir engineering applications isothermal compressibility

is often used to describe the fluid compressibility away from the wellbore. Inside the well,

due to expansion, the fluid may undergo heating or cooling and process may become

nonisothermal. If the heat loss can be neglected, the isentropic compressibility may better

represent the pressure and volume changes.

Practicing engineers in order to obtain the two-phase compressibility often use the following

relationship (Firoozabadi, 1999)

LLTGGTMT SKSKK ⋅+⋅= (5.7)

where SG and SL are the volumetric fractions of the gas and liquid components, respectively.

The equation is invalid where there is mass transfer between the phases (Firoozabadi, 1999)

what is of vital importance for water and its vapor system discussed in further section.

5.3. Compressibility of steam-water system

Calculations of the sound speed in two-components two-phase systems is an easy procedure

if adiabatic equation of state data are available to calculate compressibility, because pressure

and temperature may be considered to be independent variables in such systems. Calculation

of the sound speed in one-component two-phase system is more difficult matter because the

pressure and temperature are not independent variables and are related by the Gibbs equation

for equilibrium between both phases (Kieffer, 1977).

The complex physical process which occurs during propagation of sound wave in water-

vapor two-phase system is given in Appendix C. Propagation of the sound wave in the fluids

is accomplished by compression or rarefaction. If steam-water system remains in thermal

equilibrium where variables characterizing the system may be considered to follow the

saturation line, there must be mass transfer between the phases, since the fraction of steam in

mixture changes due to evaporation or condensation process. When the pressure wave passes

the adiabatic compression causes pressure increase in both phases; as a result the water


58

becomes subcooled, and the steam becomes superheated. The induced temperature difference

between the steam ad water phases leads to the heat transfer from superheated steam to

subcooled water (Kieffer, 1977). Depending on the original phase composition, whether

steam or water was the dominant phase heat transfer will cause water evaporation or steam

condensation. These heat and mass transfer cause that both water and steam are restored to

the saturation line. Similar process takes place in cause of pressure reduction.

In general consideration the evaporation and condensation can not take place instantaneously,

since transportation of heat and mass can only occur at a finite speed. The time period in

flashing water to steam or condensing steam to water is important in determination the degree

of equilibrium obtained in the sound wave. Since condensation and evaporation generally

proceed at different rates, it should be expected that compression and rarefaction waves

behave differently. Experiments have confirmed this theory that finite amplitude rarefaction

waves in steam water mixtures have lower velocities than compression waves because

rarefaction waves tend to maintain continuous equilibrium (McWilliam and Duggins, 1969).

A mixture of liquid and its vapor may respond to pressure disturbances by equilibrium and

nonequilibrium state. Nearly high frequencies waves the process of pressure wave

propagation follows fast and may be considered adiabatic where there is no equilibrium

between the phases. In this case the mass transfer can be neglected and thus calculations of

speed of sound are greatly simplified. The pressure pulse waves are low frequency

(Gudmundsson and Celius, 1999; Falk 1999) thus equilibrium response needs to be

considered and the mass transfer between the liquid and its vapor occurs in a time short with

comparison to the acoustic wave period.

Due to complex physical process related to the acoustic wave propagation in one-component

system different approach than for two-component need to be considered in order to find

steam-water compressibility. Experimental data on sonic velocity in steam-water system at

geothermal wellbore conditions are not readily available. To solve this problem theoretical

calculation need to be carried out to find the compressibility of this system for ideal

thermodynamics conditions. The method to derive the compressibility of a steam-water

mixture in contact with reservoir rock was elaborated by Grant and Sorey (1979). Similar


59

approach was adapted to geothermal wells by Gudmundsson et al., (1998). The studies

below are based on the work of these authors.

Two-phase fluid in geothermal well can be considered to be in thermodynamics equilibrium

where the flow is homogenous, steady-state and one dimensional. Assuming adiabatic

conditions no heat loss and gain a balance equation can be written for steam-water mixture

flowing from one infinitesimal cross section to another (Gudmundsson et al., 1998)

( ) 22221111 )1(1 LGLG hxhxhxhx ⋅−+⋅=⋅−+⋅ (5.8)

where x represents the mass fraction of the steam, and hG and hL [kJ/kg] are the enthalpy of

steam and vapor and liquid water, respectively. The equation can also be written as

222111 LGLLGL hxhhxh ⋅+=⋅+ (5.9)

the hLG [kJ/kg] is the latent heat of vaporization which may be assumed to change negligible

between adjacent cross-sections; from one infinitesimal cross-section to another. Thus the

equation (5.17) can be rewrite in form

( ) LGLL hxxhh ⋅−=+ 1221 (5.10)

Since for liquid water a change in enthalpy is equal to the heat addition at constant pressure

pp T

hC ⎟⎠⎞

⎜⎝⎛∂∂

= (5.11)

the equation (5.17) can be written as

( ) LGLp hxxTC ⋅−=∆⋅ 12 (5.12)


60

When pressure is lowering the dominant volume change is that caused by phase change;

flashing liquid into steam vapor. As water before changed phase has occupied a volume

∆m·vW, after that occupies larger volume being in gas phase ∆m·vG. This increase of volume

can be written as

( )LG vvmV −⋅∆=∆ (5.13)

The mass of water that changed phase is simply

( ) mxxm ⋅−=∆ 21 (5.14)

Therefore the change in mixture volume becomes

( ) ( )LG vvxxmV −⋅−⋅=∆ 12 (5.15)

As shown in previous section the compressibility may be defined in several ways depending

on what physical property is assumed to be constant, temperature, enthalpy or entropy. The

question arises what conditions may be found as the fluid flows in the wellbore. Usually as

defining compressibility the temperature is assumed constant, giving an isothermal

compressibility. For two-phase wellbore flow without heat loss or gain adiabatic process

seems to be most appropriate. From thermodynamics we know that an isentropic process is

both adiabatic and reversible. The flow in pipes, pipelines and wells, frictional pressure loss

makes the process non-reversible. This aspect of fluid flow is particularly important in

situation where rapid pressure drop occur for example in nozzles (Watters, 1978). It may be

considered less important in situations where the pressure changes gradually with distance for

example if wellbores and long pipelines. Since the sonic velocity of steam-vapor flowing in

the well may be approximated to isentropic conditions for which the acoustic velocity is

defined in equation (5.1).


61

Taking the compressibility from equation (5.3), and eliminating (x2-x1) from equation (5.23)

by substituting equation (5.20), the compressibility may be written in form (Gudmundsson et

al., 1998)

( )LG

LGLp

Ms hp

vvTCmV

K⋅∆

−⋅∆⋅⋅⋅−=

1 (5.16)

The fraction represents the total mixture density MMV

m ρ= defined in equation (4.7).

At all condition fluid maintain equilibrium between the phases and follows the saturation line

thus

satTp

Tp

⎟⎠⎞

⎜⎝⎛∆∆

=∆∆ (5.17)

Substituting from equation (4.7) the compressibility is

( ) ( )( )

LGsat

LGLGLps

hTp

vvCK

⋅⎟⎠⎞

⎜⎝⎛∆∆

⋅−−⋅⋅−⋅=

ραρα 1 (5.18)

satT

p⎟⎠⎞

⎜⎝⎛∆∆ may be calculated using steam tables what bring considerable inconvenience in

evaluating compressibility numerically. This can be avoided using Clausius-Clapeyron

equation

( ) ( )LG

LG

sat vvTh

Tp

−⋅+=⎟

⎠⎞

⎜⎝⎛∆∆

15.273 (5.19)


62

Compressibility expressed in this form is convenient for numerical calculations. The

thermodynamics properties values can be obtained from wellbore simulator output file and

then sonic velocity can be calculated.

5.4 Acoustic velocity models

The formulas derived for acoustic velocity base on experiments performed and varies with

respect to the components that was applied and flow patterns encountered during the

experiments. The flow regimes are described in chapter 4 of this work which deals with

multiphase flow in wells. Also the fact is important, whether homogeneity can be assumed or

not. For homogenous flow the slippage between the phases may be neglected. The study of

different models that was made here has the purpose to find the most appropriate model for

sonic velocity in one component steam-water mixture and two component gas-oil mixtures.

In present thesis acoustic velocity in steam-water mixture was calculated from Wood (1941)

equation. He derived the equation for the velocity of sound in a homogenous two component

media, based on air-water experiments (Wood, 1944). The author proposes instead of

differentiating the density of liquid mixture as shown in (5.1) relate the sonic velocity directly

to the properties of the gases and liquids. This model assumes that overall compressibility of

the mixture is related to the compressibility of the constituents by the relation

( )αα −⋅+⋅= 1LSGSMS KKK (5.20)

Substituting the compressibility of single phases from equation (5.2) yields

( )

22

1

11

LLGG

LGWood

aa

a

⋅−

+⋅

⋅−+⋅=

ρα

ρα

ραρα (5.21)


63

where α is the void fraction and ρ is the density, with the subscripts G and L indicating the gas

and liquid, respectively. Nakoryakov et al. (1993) provided that Wood’s equation can be

derived from equation (5.1) assuming the mass fraction of each phase reminded constant

what is not such a good assumption for steam-water system. Another theoretical limiting case

for vapor-liquid had the changes in density exclusively included by variation in the gas-mass

fraction due to flashing. In experiments however practically nobody has manage to observe

disturbances propagating with this flashing velocity (Falk, 1999). Semenow and Kostern

(1964) found experimentally sonic velocity in steam water flow agreeing well with Wood’s

estimate, and Noryakov et al. (1993) claimed that acoustic of Wood was the low frequency

limit for bubbly flow when the wave process were isothermal. For high frequencies the wave

process was adiabatic without energy transfer leading to the “frozen” velocity not treated here.

(Nakoryakov et al., 1993).

The dependence of propagation velocity on flow regimes demonstrates analytical expression

derived by Henry for bubbly flow using a slip flow model (Falk, 1999).

( )( ) ( )

22

1

1111

LLGG

LGHenry

aaK

KKa

⋅−

+⋅⋅

⋅−+⋅⋅−⋅−=

ρα

ρα

ραραα (5.22)

The author claimed that interfacial momentum transfer exhibited a strong influence on the

propagation velocity, and that different flow structures would have different interfacial drag

during the passage of the wave. Assuming homogenous flow with sleep K=1, equation (5.29)

reduces to Wood’s equation (5.28). However formula for acoustic velocity in slug flow

overestimates the results giving velocities close to the velocity of sound in gas phase (Falk,

1999). As the slug flow is mostly encountered flow pattern in flowing geothermal wells, this

model may bring the wrong results.

There are a few examples of analytical models for sonic velocity in gas liquid mixtures in

literature (McWilliam and Duggins, 1969; Kiefer, 1977; Firoozabadi 2000). These models


64

are based on equation (5.1) and relate the density of the mixture to the densities gases and

liquids. To make the differentiation in equation (5.1) possible, the ideal gas law was

employed for the gas and adiabatic state equation for the liquid. The derived expression for

the density of mixture was then differentiated with respect of the pressure. Henry et al. (1974)

in the book of thermodynamics properties of hydrothermal systems showed that steam

express the significant deviation from the perfect gas under high pressure conditions. Such

high pressures may be expected in geothermal wells treated in this work, thus these models

also do not occur to be the most applicable.

It is well known that properties of the natural gas also deviate from perfect gas for high

pressures, thus from the same purpose acoustic velocity in oil - gas mixture was calculated

using the formula developed by Gudmundsson and Dong (1993). The authors similarly to

Wood relate the sonic velocity directly to the properties of the gases and liquids

(Gudmundsson and Dong, 1993). Their formula presented below, was developed for gas-oil

mixture and assumes that liquid phase may contain water.

The thermodynamics compressibility and densities of mixture components are related

assuming the phases are homogenously distributed in the liquid. From equation (4.7) the

density of gas/liquid mixture is given by

( ) LGM ραραρ ⋅−+⋅= 1 (5.23)

where α is the gas/liquid void fraction, and the subscripts G and L stand for gas and liquid,

respectively. If the liquid phase contains both water and oil, the density of the liquid phase

can be similarly obtained from

( ) OWL ρβρβρ ⋅−+⋅= 1 (5.24)

where β is the water-oil volumetric fraction and the subscripts W and O stands for water and

oil, respectively.


65

Substituting (5.24) into (5.23) gives the density of gas-oil-liquid mixture

( ) ( )[ ]OWGM ρβρβαραρ ⋅−+⋅⋅−+⋅= 11 (5.25)

The relationship (5.7) can be written for: gas-oil-water mixture as

WWTOOTGGTMT SKSKSKK ⋅+⋅+⋅= (5.26)

where the S – volumetric fraction of the component is in fact the void fraction α and SO and

SW is the liquid holdup (1-α). Thus (5.26) can be alternatively written as

( ) LTGTMT KKK αα −+⋅= 1 (5.27)

If the liquid contains water and oil, its isothermal compressibility can be expressed as

( ) OTWTLT KKK ⋅−+⋅= ββ 1 (5.28)

For gas-oil-water mixture, the heat capacities at constant temperature and constant volume

may be expressed as

( ) ( ) ( )[ ]OpWpGpLpGpMp CyCyxCxCxCxC ⋅−+⋅⋅−+⋅=⋅−+⋅= 111 (5.29)

and

( ) ( ) ( )[ ]OVWVGVLVGVMV CyCyxCxCxCxC ⋅−+⋅⋅−+⋅=⋅−+⋅= 111 (5.30)


66

Where x is the gas-liquid mass fraction and y is the water-oil mass fraction. It should be noted

that mass fraction instead void fraction is employed in these equations because the ratio of

specific heats is a property based on mass unit (Gudmundsson, 1993).

From equations (5.2) and (5.6) speed of sound for mixture is given as

MTM Ka

⋅=ρ

γ2 (5.31)

where γ is the ratio of specific heats

V

p

CC

=γ (5.32)

Substituting (5.25), (5.26) and (5.32) to equation (5.31), speed of sound for gas-oil-water

mixture can be written as

( ) ( )( )[ ] ( ) ( )( )[ ]( )[ ] ( ) ( )( )[ ]OTWTGTOWG

OVWVGVOpWpGp

KKKCyCyxCxCyCyxCx

aββααρβρβαρα −+⋅−+⋅⋅⋅−+⋅⋅−+⋅

⋅−+⋅⋅−+⋅⋅−+⋅⋅−+⋅=

11)1()1(11/11

(5.33)

5.5 Attenuation mechanisms of sound wave

When sound spreads trough the media there are a diminution mechanisms of intensity or

attenuation as the distance from source increase due to loses of mostly frictional character.

All media capable of transmitting sound, are limited in extend and sooner or later the wave

must stop or change from a medium to another.

Thermal conductivity and shear viscosity attenuate the wave motion. Both the speed and

attenuation of the sound waves depend on the frequency. General considerations say that

pressure waves will pass through structures if they are of the frequency for which the


67

wavelength is larger than the structure. If the structure is a bubble or a slug in gas-liquid flow,

a pressure pulse needs to have a wavelength greater than the size of the bubble and slug to

propagate through the gas-liquid flow (Gudmundsson and Celius 1999).

In multiphase flow there are additional attenuation mechanisms (Falk, 1999) which include:

- viscous drag

- steady interfacial drag

- added mass (interior effect)

- boundary layer around a particle/drop (Basset force)

- interfacial heat exchange

- compressibility of each phase

- concentration gradient effects

- phase transitions

- deformation and fragmentation of bubbles and drops

- reflections at the interface

Knowledge of which of these attenuation mechanisms are the most important can greatly

simplify the models for sound wave propagation. The relative importance, however, depends

on the media, flow pattern and the frequency.

In approach towards zero frequency where all processes take place slowly, the compression

and expansion of the fluid occur reversibly and adiabatically. Trusler (1991) claimed that

boundary-layer absorption was the most important attenuation mechanism for low-frequency

waves in tubes with single phase flow. However, at high frequencies the fluid cannot

maintain local equilibrium. Consequently, some of the energy is dissipated and high

frequency waves will be highly attenuated by reflections of the interfaces and can not

propagate very far in bubbly and slug flow (Falk, 1999). The dominant frequency in low-

pressure air-water flow in pipelines has been shown to be in the range 1-10 Hz (Dong and

Gudmundsson 1993). Similar results were reported by Falk (1998). Therefore, pressure


68

waves in gas-liquid flow are infrasonic (lower frequency than audible sound which is about

20,000Hz).

Numerical calculations in bubbly flow showed that the slippage between the phases has little

influence on the wave structure (Falk, 1999), though it can only be neglected when the liquid

viscosity is considerable higher than of water, or if the bubbles are small. In bubbly flow the

main attenuation mechanism was heat exchange between the gas bubbles and surrounding

liquid. Kiefer (1977) showed addition theory that added liquid mass is important for the

pressure wave propagation. Calculations of speed of sound in bubbly flow (McWilliam and

Duggins, 1970) and (Kiefer, 1977) showed that surface tension is important for small bubbles,

while at large bubbles only liquid compressibility is important. Firoozabadi (2000) also

revealed that that capillary pressure may affect the two-phase compressibility only in porous

media at reservoir conditions outside the wellbore where phases interface is curved.

5.6 Concluding remarks

1. The acoustic velocity in steam-water two phase systems is more complicated than for two-

component systems. In two-component system gas being dissolved under high pressure

comes out of solution as the pressure decrease and is characterized by solution gas-oil ratio Rs.

In one component system gas emerges due to evaporation process and if the pressure change

is not of the high frequency the equilibrium is maintained between the phases. As the system

responses with equilibrium for pressure change the temperature effects and mass transfer

between both phases is essential for calculations.

2. The study of models made in this chapter showed that there is not one best model for

acoustic velocity in one component steam-water mixture. The Wood equation was chosen for

calculations. The choice of this model is mainly due to Semenow and Kostern (1964)

experiments that showed a sound velocity in steam water flow agreeing well with those

estimated from Wood’s model. However the other authors show the limitation of this model

in many areas. The speed of sound in oil gas mixtures was calculated from formula


69

Developed by Gudmundsson and Dong (1993). This formula is reported by authors to be in

good agreement with measured data.

Both chosen models require the properties of each constituent phase to be known. The

procedure necessary in order to calculate these properties was fully described in this chapter

for both gas-oil and steam-water cases. For oil and gas two-phase flow thermodynamics

compressibility can be readily obtained from simulations performed. Unfortunately software

available for geothermal simulations is not such sophisticated as those available for

petroleum applications and thus steam compressibility need to be calculated from equation

derived in this chapter.

3. The acoustic velocity in liquid single-phase is higher from acoustic velocity in pure gas

phase due to significantly lower compressibility of liquid phase. At the presence of only one

percent by volume gas in the form of gas bubbles the acoustic velocity decreases dramatically

and two-phase system presents different character from each of the constituent phases. The

explanation is the fact that such two-phase system has density of liquid and compressibility

of gas. The calculated acoustic velocities for gas-oil and steam-water mixtures for wide range

of void fraction values are presented on Figure 5.1 and Figure 5.2. The plots also shows that

depression in the sonic velocity is less if the pressure is increased while retaining the same

void fraction, although the sensitivity to change in pressure becomes progressively

diminished.

4. The plots of acoustic velocity vs. void fraction for oil-gas and steam-water systems

revealed different behavior for high void fraction values. For hydrocarbons two-phase fluids

mixture the plot has less slope which becomes more flat as the pressure is higher. For steam-

water mixture this shape is more inclined towards vertical. The calculations made showed out

that this effect may be important for lower pressure values where water dryness has higher

values and steam occupies main volume of the pipe. This high value void fraction changes

has more effect on acoustic velocity than pressure changes and thus acoustic velocity

decrease with pressure increase across the wellbore. This problem will be presented precisely

together with calculations made in Chapter 8.


70

5. Gudmundsson and Dong model relate the sonic velocity directly to the properties of the

gases and liquids in the same manner as Wood’s model. In fact assuming that mixture do no

contain water and β = 0, the equation may be expressed in the same form as Wood’s. The

difference is in the approach towards the thermodynamic process that occurs during the

sound propagation.

Compressibility is often defined as the small volume change than occur in closed system at

constant temperature. The second law of the thermodynamics tells that sound propagation

process must be isentropic due to there is no temperature gradient except the wave itself.

Wood’s proposed to calculate speed of sound separately for both phases from isentropic

compressibility. Gudmundsson and Dong equation use the isothermal compressibility and

then transform the equation to the isotropic condition using the specific heats ratio as shown

in equation (5.31). The possible error due to assuming isothermal process instead isentropic is

up to 7% for gas-oil mixtures and even up to 14% for steam-water depending on the pressure.

Table 5.1 contains calculated results for pressure equal 52.4 bar, for gas-oil mixture and

Table 5.2 shows similar calculations for steam-water mixture at the 45.1 bar. The differences

between calculated sound speed values assuming whether isothermal or isentropic process are

also shown in form of plots by Figure 5.3 and 5.4 for gas-oil and steam-water respectively.


71

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

acou

stic

vel

ocity

[m/s

]

p = 100.47

p = 52.36

p = 19.45

Figure 5.1 Calculated acoustic velocities Vs void fraction for oil-gas mixture

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

acou

stic

vel

ocity

[m/s

]

p = 45.2 [bar]p = 30 [bar]p = 15 [bar]

Figure 5.2 Calculated acoustic velocities Vs void fraction for steam-water mixture


72

Table 5.1 Calculated acoustic velocities for gas-oil mixture.

oil-gas mixture, p = 70.2 [bar]

α a [m/s]

isentropic

a [m/s]

isothermal difference [m/s] %

0.1 268.3 250.8 17.5 6.8

0.3 176.4 164.8 11.6 6.7

0.5 158.5 148.5 10.0 6.5

0.8 178.4 169.0 9.4 5.4

0.9 207.0 198.6 8.4 4.1

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

acou

stic

vel

ocity

[m/s

]

isentripic p = 45.1 [bar]

isothermal p = 45.1 [bar]

isentropic p = 70.2 [bar]


Figure 5.3 Calculated acoustic velocity for gas-oil mixture assuming isothermal and

isentropic process of sound propagation


73

Table 5.2 Calculated acoustic velocity for steam-water mixture.

steam-water mixture, p = 19.4 [bar]

α a [m/s]

isentropic

a [m/s]

isothermal difference [m/s] %

0.1 282.7 247.1 35.6 13.4

0.3 187.3 163.2 24.1 13.8

0.5 170.9 148.8 22.1 13.8

0.8 205.5 179.6 25.9 13.5

0.9 258.0 226.8 31.2 12.9

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]




isothermal p =52.4 [bar]

Figure 5.4 Calculated acoustic velocity for steam-water mixture assuming isothermal and

isentropic process of sound propagation

Chapter 6

Case Studies

6. Case Studies __________________________________________________________________________

75

6.1 Calculation purpose

The purpose of these calculations is to estimate water-hammer and line packing effects when

valve has been fully closed. The results from these calculations should give a picture of these

phenomena in two-phase wells. Then similar calculations for oil and geothermal wells may

be compared witch each other. A description of the effect should give a result which can be

linked to field measurements to see if any other factors might affect the pressure gradient

after the valve closure. Offshore Pressure Pulse tests have been made in several North Sea

wells to validate the theoretical simulation. The tests issues were described in two

confidential reports and then published (Gudmundsson and Falk, 1999), (Gudmundsson and

Celius, 1999). Geothermal wells are planed to be tested for Pressure Pulse method in summer

2003.

The water-hammer effect is caused by rapid closure a valve in flowing pipe. The pressure

increase is dependant on fluid density, velocity and sonic velocity in the flowing well as

shown in Chapter 3. These will be calculated here. In liquid flow, line packing is the increase

in pressure caused by a wall friction when a valve is closed and the fluid is stopped by a

pressure wave which is emitted from the valve in the instant of closure. In liquid/gas two-

phase flow, the increase in pressure is the sum of pressure drop due to wall friction and the

pressure drop due to interfacial friction. The interfacial friction is an unknown quantity, while

the frictional pressure drop can be considered a known quantity.

6. Case Studies – oil wells __________________________________________________________________________

76

6.2 Water-hammer and line packing in oil wells

The three programs used were PipeSim 2000, GOW 3.0 and Excel. PipeSim is a multiphase

flow simulator. The futures and modes of the program used in present work are described in

Appendix D. PipeSim simulates the flow of an oil/gas/water mixture in a well or pipeline.

The program takes the well fluid data and uses them to calculate various properties of the

fluid and system. PipeSim was also used for inflow performance (IP) and tubing performance

(VLP) in order to estimate the flowrates for different wellhead pressures.

The GOW program is published by Gulf Publishing Co. Houston, Texas. It allows calculating

the various parameters of many substances including oil, gas and water. The PipeSim output

file does not give the values of oil and gas compressibility and these were found from this

program for given PVT conditions taken from well profile.

The Excel spreadsheet was used in present work for additional calculations of acoustic

velocity. The acoustic velocity was calculated using formula reported by Dong and

Gudmundsson (1993) and given in equation (5.40) in this work. Excel was also used to plot

the results.

The above programs may be substituted by similar programs. Prosper Multiphase Flow

Simulator, designed by Petroleum Experts Ltd. Edinburgh gives similar output files suitable

to present calculations. Only the PipeSim was available at university’s computers thus

calculations placed in this thesis work hail from PipeSim. Excel may be replaced by almost

any spreadsheet that can plot and calculate. Also GOW has equivalent programs.

To run PipeSim the fluid, completion and production data are necessary. The data used in

these simulations for one well was obtained from work of Jonsson (1995). This contains the

data about production, completion and fluid properties from the Draugen oil field. Two other

wells were simulated based on data from work of Falk (1999). These data were gained during

the Pressure Pulse Method tests on Gullfaks and Oseberg platforms. In addition some


77

reference values for these oil fields were taken from the Skjæveland and Kleppe Monograph

(1992) that contains the characteristics of the most North Sea fields.

PipeSim was used to determine the properties of the oil and gas at different depths in the

producing well. Simulations gave the values at 50 meter intervals. Oil pressure, temperature,

density, void fraction, gas and oil heat capacities and compressibility are the properties

required to calculate the acoustic velocity. The last two were computed in GOW and then

entered into the Excel spreadsheet. The void fraction is also not given directly in PipeSim

output file but as water cut is 0 (assumed in all simulations done), the void fraction and the

liquid holdup add to one.

Excel was used to plot the results. The depth, acoustic velocity and the pressure drop due to

friction were entered into spreadsheet. The frictional pressure drop is available on the

PipeSim output file. These values are then used to calculate travelling time for a pressure

pulse down to a certain point and up again. Calculated results were plotted against each other.

Depth and time are plotted on the x axis and total pressure drop and acoustic velocity on the y

axis. The total pressure drop versus time gives the effect of line packing.

PipeSim gives the output file compatible with excel format thus the results are presented in

two forms; Excel spreadsheet and Excel plots. Excel spreadsheet contains the calculated

values of various properties affecting the acoustic velocity and line packing listed in columns

and is available from all simulated oil wells in Appendix F of this work. These values were

directly used to plot the results.

Table 6.1 contains the data about the wells that was chosen to illustrate the pressure pulse

method in oil wells. These data include well depth, tubing inner diameters, SCSSV depth and

inner diameters, reservoirs pressures and temperatures necessary to run simulations in

PipeSim. Table 6.2 contains the molecular composition of the well fluids used in simulations.

PipeSim base on the reservoir data to predicts the well flowing pressure at the bottom of the

well and then simulate the flow in vertical well. Table 6.3 presents the calculated parameters.

The wellhead pressures, temperatures and flowrates are presented in the table together with

parameters that make up water-hammer: density, acoustic velocity and flow velocity.


78

Table 6.1 Well geometry and reservoir data for computer simulations

WELL SYMBOL A1 A2 B C

Depth H (m) 1893 1893 1952 1924

Tubing ID (inch) 6.184 6.184 4.5 5.125

SCSSV depth (m) 650 650 none none

SCSSV ID (inch) 5.963 5.963 none none

Reservoir pressure pres (bar) 165 165 254 314

Reservoir temperature Tres (oC) 71 71 73 75

Table 6.2 Molecular composition of the well fluids (measured as a mole fraction of the gas

phase at stock-tank conditions)


N2 0.090 0.090 0.307 0.320

CO2 0.280 0.280 0.996 0.620

C1 46.49 46.49 46.323 44.63

C2 6.14 6.14 4.045 3.83

C3 4.60 4.60 0.881 0.94

iC4 0.92 0.92 0.556 0.57

nC4 2.31 2.31 0.511 0.56

iC5 0.99 0.99 0.662 0.6

nC5 1.35 1.35 0.293 0.28

C6 2.04 2.04 1.014 0.92

C7 3.15 3.15 2.869 2.28

C8 3.35 3.35 4.064 4.05

C9 2.18 2.18 3.257 3.36

C10+ 25.54 25.54 34.222 37.04


79

Table 6.3 Calculated values at the inlet and wellhead condition


Inlet pressure - pwf (bar) 156 158.5 258.7 228.97

Inlet temperature – Twf (oC) 71 71.0 72.0 72.85

Stock tank oil flow rate - QL (Sm3/day) 7300.0 5125.0 2165 1024.9

Stock tank gas flow rate - QG (million m3/day) 0.3796 0.26651 0.20352 0.0641

Total mass flowrate - m (kg/s) 75.144 52.755 22.708 10.794

Wellhead pressure pwh (bar) 8.95 20.35 123.5 65.12

Void fraction – α 0.827 0.601 0.210 0.252

Mixture density– ρM (kg/m3) 142.5 322.4 616.2 623.5

Fluid mean velocity – u (m/s) 18.438 8.782 2.906 1.607

Acoustic velocity a (m/s) 87.7 102.7 288.4 192.9

Water-hammer ∆pa (bar) 2.30 2.91 5.16 1.93

Figure 6.1 contains two plots that present estimated water-hammer and line packing for oil

wells taken into considerations in present work. The first pressure increase on the plots is the

water hammer effect after valve closure, and then the long line packing shows the pressure

build up due to friction. The rapid pressure increase in line packing starts at the time where

bubble point is reached.


80

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

time elapsed [s]

pres

sure

[bar

]

Well A1 Well A2

Well C

Well B

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

25.0

27.5

30.0

32.5

35.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

time elapsed [s]

pres

sure

[bar

]

Well B

Well A1 Well A2

Well C

Figure 6.1 Estimated water-hammer and line packing in oil wells


81

Calculated results are presented for each well in the form of plots. The list of plots is given

below:

mixture density Vs depth, …………………………………….….... Figure 6.2

void fraction Vs depth………………………………………………. Figure 6.3

densities ratio Vs velocities ratio……………………………......….. Figure 6.4

logarithm of densities ratio Vs logarithm of velocities ratio.............. Figure 6.5

acoustic velocity Vs depth……………………………………….…. Figure 6.6

acoustic velocity Vs time………………………………...…………. Figure 6.7

sum frictional pressure drop Vs depth……………...…..…...……… Figure 6.8

sum frictional pressure drop Vs time……………………..……….... Figure 6.9

pressure (line packing) Vs time……………………………..…..….. Figure 6.10

acoustic velocity Vs void fraction (for increasing pressures)......…... Figure 6.11

elevation Vs pressure……………………………………….....…..... Figure 6.12

The depth is the depth of the point in question below the wellhead. The time elapsed is the

time period it takes for a sound wave to travel to the point in question from the wellhead and

back again, i.e. down and up. The sum frictional pressure drop is the total pressure drop

between the point in question and the wellhead.

6. Case Studies – well A1 __________________________________________________________________________

82

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.2 – A1, mixture density fraction Vs depth

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

void

frac

tion

[%]

Figure 6.3 – A1, void fraction Vs depth


83

0

10

20

30

40

50

60

70

80

90

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4velocities ratio (slip) vG/vL

dens

ities

ratio

DL/

DG

bubble II

bubble I

liquid only

Transition of f low pattern f low revealed on this plot, it w as not reported in simulator output f ile.

Figure 6.4 – A1, densities ratio Vs velocities ratio

1

10

0.0 0.1 1.0velocities ratio (slip) log(vG/vL)

dens

ities

ratio

log(

DL/

DG)

bubble I

bubble II

Figure 6.5 – A1, logarithm of densities ratio Vs logarithm velocities ratio


84

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

acou

stic

vel

ocity

[m/s

]

Figure 6.6 – A1, acoustic velocity Vs depth

0

200

400

600

800

1000

1200

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15time [s]

acou

stic

vel

ocity

[m/s

]

Figure 6.7 – A1, acoustic velocity Vs time


85

0

2

4

6

8

10

12

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]

Figure 6.8 – A1, sum of frictional pressure drop Vs depth

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

time [s]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]


86

Figure 6.9 – A1, sum of frictional pressure drop Vs time

0

2

4

6

8

10

12

14

16

18

20

22

24

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18time elapsed [s]

pres

sure

[bar

]

0

1

1

2

2

3

3

4

4

5

5

time differential dp/dt

line packing

time differential

Figure 6.10 – A1, line packing and time differential


87

Calculated acoustic velocities for subsequent pressures across the wellbore

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9void fraction

soni

c ve

loci

ty [m

/s]

750m

600m

450m

300m

150m

0m

path

Figure 6.11 – A1, acoustic velocity Vs void fraction (for increasing pressure)

Figure 6.12 – A1, elevation Vs pressure


88

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

mix

ture

den

sity

[kg/

m3]


0

10

20

30

40

50

60

70

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

void

frac

tion

[%]



89

0

5

10

15

20

25

30

35

40

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4velocities ratio (slip) vG/vL

dens

ities

ratio

DL/

DG

Transition betw een liquid, bubble and slug f low

bubble

slug

liquid only


1

10

0.0 0.1 1.0velocities ratio (slip) log(vG/vL)

dens

ities

ratio

log(

DL/

DG

)



90

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

acou

stic

vel

ocity

[m/s

]


0

200

400

600

800

1000

1200

0 1 2 3 4 5 6 7 8 9 10 11 12

time [s]

acou

stic

vel

ocity

[m/s

]



91

0

2

4

6

8

10

12

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]


0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9 10 11 12

time [s]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]



92

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

time elapsed [s]

pres

sure

[bar

]

0

1

2

3

4

5

6

7


line packing

time differential



93

Calculated sonic velocities

80

130

180

230

280

330

380

430

480

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9void fraction

soni

c ve

loci

ty [m

/s]

600m

500m

400m

300m

200m

100m

0m

path



6. Case Studies – well B __________________________________________________________________________

94

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200 1400 1600 1800 2000depth [m]

mix

ture

den

sity

[%]

Figure 6.2 – B, mixture density fraction Vs depth

0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

void

frac

tion

[%]

Figure 6.3 – B, void fraction Vs depth


95

0

1

2

3

4

5

6

7

8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6velocities ratio (slip) vG/vL

dens

ities

ratio

DL/

DG

liquid only

bubble f low

Figure 6.4 – B, densities ratio Vs velocities ratio

0

10.1 1.0

velocities ratio (slip) log(vs/vw)

dens

ities

ratio

log(

Dw

/Ds)

.

Figure 6.5 – B, logarithm of densities ratio Vs logarithm velocities ratio


96

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

acou

stic

vel

ocity

[m/s

]

Figure 6.6 – B, acoustic velocity Vs depth

0

200

400

600

800

1000

1200

1400

0 1 2 3 4 5 6 7 8

time [s]

acou

stic

vel

ocity

[m/s

]

Figure 6.7 – B, acoustic velocity Vs time


97

0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]

Figure 6.8 – B, sum of frictional pressure drop Vs depth

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8

time [s]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]

Figure 6.9 – B, sum of frictional pressure drop Vs time


98

60

70

80

90

100

110

120

130

140

150

160

170

180

190

0 1 2 3 4 5 6 7 8 9 10time elapsed [s]

pres

sure

[bar

]

0

2

4

6

8

10

12


line packing

time differential

bubble point pressure

Figure 6.10 – B, line packing and time differential


99


0

200

400

600

800

1000

1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

1700m 1400m 1100m 800m 500m 200m 0mpath

Figure 6.11 – B, acoustic velocity Vs void fraction (for increasing pressure)

Figure 6.12 – B, elevation Vs pressure

6. Case Studies – well C __________________________________________________________________________

100

0

100

200

300

400

500

600

700

800

900

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

mix

ture

den

sity

[kg/

m3]


0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

void

frac

tion

[%]



101

0

2

4

6

8

10

12

14

16

18

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6velocities ratio (slip) vG/vL

dens

ities

ratio

DL/

DG

bubble

liquid only


0.1

1.0

10.0

0.000001 0.000010 0.000100 0.001000 0.010000 0.100000 1.000000velocities ratio (slip) log(vG/vL)

dens

ities

ratio

log(

DL/

DG

)

f low transition bubble



102

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

acou

stic

vel

ocity

[m/s

]


0

200

400

600

800

1000

1200

0 1 2 3 4 5 6 7 8 9

time [s]

acou

stic

vel

ocity

[m/s

]



103

0

1

2

3

4

5

6

0 200 400 600 800 1000 1200 1400 1600 1800 2000

depth [m]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]


0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9

time [s]

sum

fric

tiona

l pre

ssur

e dr

op [b

ar]



104

64

65

66

67

68

69

70

71

72

73

74

0 1 2 3 4 5 6 7 8 9 10 11

time elapsed [s]

pres

sure

[bar

]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5


line packing

time differential



105


100

200

300

400

500

600

700

800

900

1000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

1050m 900m 700m 500m 300m 100m 0mpath



6. Case Studies – geothermal wells __________________________________________________________________________

106

6.3 Water-hammer and line packing in geothermal wells

Two programs used in this case were HOLA 3.1 wellbore simulator, and Excel (also

MATHLAB 6.0 and GOW 3.0 but in limited single tasks). The features and modes of HOLA

program that was used in present work are described in Appendix E. The program requires

the well and feedzone parameters in order to calculate the various parameters of fluid and

system in wellbore. This program was also used for inflow performance (IP) and tubing

performance (VLP) purposes in order to estimate the flow parameters for different tubing size

(13⅜ and 9⅝”). The wellhead pressure was assumed to have the same value for both tubing

diameters, thus the mass flowrate must be different. Calculation confirmed that increase in

tubing size from 9⅝” to 13⅜ can bring the almost double output what justify economically

drilling such wide wells. The 13⅜” wide configuration is widely adopted in Iceland but

literature shows that 9⅝” may be considered as typical pipe diameter for high enthalpy liquid

dominated geothermal wells worldwide. That was the purpose of including the both tubing

sizes in simulations. The IP (inflow performance) curves could not be calculated directly

from HOLA and some special approach was used that is described in Appendix E. The Excel

spreadsheet was used to perform the calculations and plot the results.

The data for simulations were assumed base on information about two Iceland fields -

Reykjanes and Svartsengi, (Gudmundsson, 2003). The data include information about

average reservoir pressure and temperature, well and producing fractures depth, casing

program and wellhead production parameters. The additional like enthalpies and

productivity-indexes was computed using HOLA simulator. The wells D and E are

representing the same field thus input parameters and simulated results are similar. The wells

indicated with F are based on data from a different field thus properties and results may be

expected to vary from the previous wells.

The HOLA simulator gives the limited number of fluid properties as compared to PipeSim,

but those obtained were sufficient to continue the necessary calculations in Excel spreadsheet.

HOLA output text file gives calculated well profile containing values of pressure,

temperature, enthalpy, dryness, density, and velocity of each phase separately. The steam


107

void fraction was calculated from equation (4.5), and then the mixture density, viscosity and

enthalpy was computed from relations (4.7), (4.8), (4.9) respectively.

Acoustic velocity was obtained from Wood’s formula given as (5.21) in present work. In

single liquid phase flow sonic velocity depends mostly on water compressibility whether in

two phase region the sonic velocity is highly dependant on steam compressibility and water

compressibility has much less impact.

The steam properties in geothermal well follows the saturation line as the system tends to

remain in equilibrium between both phases. These properties are given is steam tables, what

is obviously inconvenient for computer calculations. Michaelides (1981) proposed the

polynomial expressions to solve this inconvenience problem. In present work CalcSoft 3.0

the shareware program written by M.L. McGuire was used in computations of steam and

water heat capacities at constant pressure and volume. Isothermal compressibility KTW of the

water phase was possible to compute from GOW program and the isothermal compressibility

of steam fraction was calculated from formula suitable for computer calculations, derived in

chapter 5 and finally given by the equation (5.24). The acoustic velocity allows then calculate

the travelling time of pressure pulse down to a certain point and back again.

Frictional pressure drop across the wellbore is the required value for line packing calculations.

From HOLA program only total pressure drop across the wellbore is available, thus the

friction factor f was computed for given pipe roughness from empirical formula proposed by

Haalad (Stetfjerding, 1998)

n

sn

dk

nf

⋅

⎟⎠

⎞⎜⎝

⎛⋅

+⎟⎠⎞

⎜⎝⎛⋅−=

11.1

75.3Re9.6log8.11 (6.1)

where n = 1 for gradual transition between the smooth and rough flow, or n = 3 that is

suitable for gas pipelines. The Reynolds number is given by the equitation (4.26), ks is the

wall roughness and d pipe diameter. Pressure drop due to friction may be calculated from


108

Darcy-Weisbach equation given as (2.6). The total pressure drop versus time gives the effect

of line packing.

Calculated values were then plotted in Excel. Depth and time are plotted on the x axis and

total pressure drop and acoustic velocity on the y axis. The results are presented it two forms

Excel spreadsheet and Excel plots. Excel spreadsheet contains the calculated values of

various properties affecting the acoustic velocity and line packing listed in columns and is

available in Appendix G for all simulated geothermal wells. These values were directly used

to plot the results.

Table 6.4 contains geothermal wells data used in present calculations. The table include well

depths, reservoir parameters and two stage tubing diameters. Calculated parameters are

presented in Table 6.5 which includes wellhead conditions and parameters that make up

water-hammer at the wellhead.

Estimated water-hammer and line packing in geothermal wells are shown in the form of plots.

The Figure 6.13 presents the results for the wells taken into consideration in present work, the

water-hammer pressure increase starts from the actual calculated wellhead pressure. Line

packing in the case of geothermal wells contains both bubble point and diameter change

effects. These effects are discussed precisely in the next chapter. Figure 6.14 shows the same

plots but the pressure on the y axis is the only pressure increase due to water-hammer and

friction effects, beginning from the 0 pressure value. Figure 6.15 presents only the water-

hammer pressure increase after a valve closure.


109

Table 6.4 Reservoir and geometry input data for computer simulations

WELL D1 D2 E1 E2 F1 F2

Depth (m) 2028 2028 2228 2228 1450 1450

Reservoir Pressure pr (bar) 197.8 197.8 198 198 100 100

Reservoir Temperature Tr (oC) 309.6 309.6 309.7 309.7 263.1 263.1

Downhole pressure pwf (bar) 176.3 179.3 174.3 183.7 83.1 90.2

Enthalpy at the feedzone h (kJ/kg) 1389 1389 1390 1284 1150 1129

Productivity Index PI (E-12·kg/s/m3) 2.39 2.39 1.25 1.25 3.61 3.61

Production casing ID (inch) 13 ⅜ 9 ⅝ 13 ⅜ 9 ⅝ 13 ⅜ 9 ⅝

Slotted liner ID (inch) 9 ⅝ 7 9 ⅝ 7 9 ⅝ 7

Liner from depth (m) 750 750 750 750 750 750

Table 6.5 Calculated values at the wellhead conditions

WELL D1 D2 E1 E2 F1 F2

Wellhead pressure pwh (bar) 45.17 44.7 31.94 31.93 15.02 15.06

Wellhead temperature Twh (oC) 257.6 257.0 237.3 237.3 198.3 198.5

Wellhead enthalpy hwh (kJ/kg) 1275.7 1255.4 1198.7 1168.9 1082.1 1051.0

Total mass flow m (kg/s) 42.61 25.98 49.87 30.28 47.15 28.46

Void fraction α 0.615 0.628 0.640 0.722 0.878 0.879

Fluid mean velocity (m/s) 1.48 1.18 1.80 2.21 4.61 5.46

Mixture density (kg/m3) 317.1 307.6 304.4 239.1 112.6 111.4

Acoustic velocity (m/s) 152.6 152.6 128.1 136.0 123.4 123.8

Water-hammer ∆pa (bar) 0.716 0.848 0.703 0.880 0.641 0.754


110

10

13

16

19

22

25

28

31

34

37

40

43

46

49

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19time elapsed [s]

pres

sure

[bar

]

Well D1 Well D2

Well E1

Well F1

Well E2

Well F2

Figure 6.13 Estimated water-hammer and line packing in geothermal wells


111

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

time elapsed [s]

pres

sure

[bar

]

Well E1

Well D1

Well D2

Well E2

Well F1

Well F2

Figure 6.14 Estimated water-hammer and line packing in geothermal wells


112

0.00

0.11

0.22

0.33

0.44

0.55

0.66

0.77

0.88

0.99

1.10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8time elapsed [s]

wat

er -

ham

mer

pre

ssur

e [b

ar]

Well D1Well D2Well E1Well E2Well F1Well F2

Figure 6.15 Estimated water-hammer magnitudes in geothermal wells

(valve is assumed to close completely in 0.5 s)


113

Calculated results are presented for each well in the form of plots. The list of plots is given

below:

VLP deliverability curves………………………………………...… Figure 6.16

pressure and temperature well profile…………………………......... Figure 6.17

mixture density Vs depth…………………………………………….Figure 6.18

void fraction Vs depth…………………………………………….… Figure 6.19

densities ratio Vs velocities ratio………………………………….... Figure 6.20

logarithm of densities ratio Vs logarithm of velocities ratio……….. Figure 6.21

acoustic velocity Vs depth………………………………………….. Figure 6.22

acoustic velocity Vs time ……………………………………….….. Figure 6.23

frictional pressure drop per 10 (m) Vs depth……………………….. Figure 6.24

sum frictional pressure drop Vs depth……………………………… Figure 6.25

sum frictional pressure drop Vs time……………………………….. Figure 6.26

pressure (line packing) Vs time ……………………………………. Figure 6.27

acoustic velocity Vs void fraction (for increasing pressures) …….... Figure 6.28

acoustic velocity Vs void fraction (for increasing pressures),

scaled up plot (only F1 and F2 wells) …………………………….... Figure 6.29

The depth is the depth of the point in question below the wellhead. The time elapsed is the

time period it takes for a sound wave to travel to the point in question from the wellhead and

back again, i.e. down and up. The frictional pressure drop per 10 meters is the pressure loss

due to wall friction at this distance. The sum frictional pressure drop is the total pressure

drop between the point in question and the wellhead.

6. Case Studies – well D1 __________________________________________________________________________

114

020406080

100120140160180200220240260280300320

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

m [kg/s]

Pwf [

bar]

95/8 ''

133/8 ''

Figure 6.16 – D1, well inflow and tubing performance

Figure 6.17 – D1, simulated well pressure and temperature profile


115

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.18 – D1, mixture density Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

void

frac

tion

Figure 6.19 – D1, void fraction Vs well depth


116

0

5

10

15

20

25

30

35

40

0.0 0.5 1.0 1.5 2.0 2.5velocities ratio (slip) vs/vw

dens

ities

ratio

Dw

/Ds

Figure 6.20 – D1, densities ratio Vs velocities ratio

1

10

0.0 0.1 1.0velocities ratio (slip) log(vs/vw)

dens

ities

rat

io lo

g(D

w/D

s)

Transition between bubble and slug flow

Figure 6.21 – D1, logarithm of densities ratio Vs logarithm of velocities ratio


117

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

acus

tic v

eloc

ity[m

/s]

Figure 6.22 – D1, acoustic velocity Vs well depth

0

200

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10time [s]

acus

tic v

eloc

ity[m

/s]

Figure 6.23 – D1, acoustic velocity Vs time


118

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

fric

tiona

l pre

ssur

e dr

op p

er 1

0m

[bar

/m]

diameter change discontinuity from 133/8 to 95/8

f irst bubble appears

Figure 6.24 – D1, frictional pressure drop at 10 m distance Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

/m]

Figure 6.25 – D1, sum of frictional pressure drop Vs well depth


119

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 1 2 3 4 5 6 7 8 9 10time [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

]

Figure 6.26 – D1, sum of frictional pressure drop Vs time

45.0

45.2

45.4

45.6

45.8

46.0

46.2

46.4

46.6

46.8

0 1 2 3 4 5 6 7 8 9 10 11 12time [m]

Pres

sure

[bar

]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


line packing

time differential

Figure 6.27 – D1, estimated water – hammer and line packing at the wellhead,

(together with time differentialdtdp )


120

130

170

210

250

290

330

370

410

450

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

acou

stic

vel

ocity

[m/s

]

0 m100 m200 m300 m400 m500 m600 m640 m660 mpath

Figure 6.28 – D1, Calculated sonic velocities for varying pressure across the well depth Plot presents void fraction Vs acoustic velocity changes across the well depth.

Sudden change in void fraction is caused by computational discontinuity at the 230m depth


121

020406080

100120140160180200220240260280300320

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

m [kg/s]

Pwf [

bar]

95/8 ''

133/8 ''

Figure 6.16 – D2, well inflow and tubing performance

Figure 6.17 – D2, simulated well pressure and temperature profile


122

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.18 – D2, mixture density Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

void

frac

tion

Figure 6.19 – D2, void fraction Vs well depth


123

0

5

10

15

20

25

30

35

40


dens

ities

ratio

Dw

/Ds

Figure 6.20 – D2, densities ratio Vs velocities ratio

1

10


dens

ities

rat

io lo

g(D

w/D

s)

Transition between bubble and slug flow

Figure 6.21 – D2, logarithm of densities ratio Vs logarithm of velocities ratio


124

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

acus

tic v

eloc

ity[m

/s]

Figure 6.22 – D2, acoustic velocity Vs well depth

0

200

400

600

800

1000

1200

1400

1600

0.0 2.0 4.0 6.0 8.0 10.0time [s]

acus

tic v

eloc

ity[m

/s]

Figure 6.23 – D2, acoustic velocity Vs time


125

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

fric

tiona

l pre

ssur

e dr

op p

er 1

0m

[bar

/m]

diameter change from9 3/8'' to 7''


Figure 6.24 – D2, frictional pressure drop at 10 m distance Vs well depth

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

/m]

Figure 6.25 – D2, sum of frictional pressure drop Vs well depth


126

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 1 2 3 4 5 6 7 8 9 10time [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

]

Figure 6.26 – D2, sum of frictional pressure drop Vs time

44.5

45.0

45.5

46.0

46.5

47.0

47.5

0 1 2 3 4 5 6 7 8 9 10 11time [m]

Pres

sure

[bar

]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


line packing

time differential

Figure 6.27 – D2, estimated water – hammer and line packing at the wellhead



127

130

170

210

250

290

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

0 m100 m200 m300 m400 m500 m550 m570 m590 mpath

Figure 6.28 – D2, calculated sonic velocities for varying pressure across the well depth

6. Case Studies – well E1 __________________________________________________________________________

128

0

2040

6080

100120

140160

180200

220240

260280

300

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

m [kg/s]

Pwf [

bar]

9 5/8 '' 13 3/8 ''

Figure 6.16 – E1, well inflow and tubing performance

Figure 6.17 – E1, simulated well pressure and temperature profile


129

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.18 – E1, mixture density Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.19 – E1, void fraction Vs well depth


130

0

10

20

30

40

50

60

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5velocities ratio (slip) vs/vw

dens

ities

ratio

Dw

/Ds

Figure 6.20 – E1, densities ratio Vs velocities ratio

1

10


dens

ities

rat

io lo

g(D

w/D

s)

Transition between the bubble flow and slug flow(discountinuity at the flow patterns boundary)

Figure 6.21 – E1, logarithm of densities ratio Vs logarithm of velocities


131

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

acus

tic v

eloc

ity[m

/s]

Figure 6.22 – E1, acoustic velocity Vs well depth

0

200

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10 11 12 13time [s]

acus

tic v

eloc

ity[m

/s]

Figure 6.23 – E1, acoustic velocity Vs time


132

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

fric

tiona

l pre

ssur

e dr

op p

er 1

0m

[bar

/m]


diameter change from 13 3/8" to 9 5/8"

Figure 6.24 – E1, frictional pressure drop at 10 m distance Vs well depth

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400well depth [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

/m]

Figure 6.25 – E1, sum of frictional pressure drop Vs well depth


133

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 1 2 3 4 5 6 7 8 9 10 11 12 13time [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

]

Figure 6.26 – E1, sum of frictional pressure drop Vs time

31.5

32.0

32.5

33.0

33.5

34.0

34.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15time [m]

Pres

sure

[bar

]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6


line packing


Figure 6.27 – E1, estimated water – hammer and line packing at the wellhead



134

110

150

190

230

270

310

350

390

430

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

0 m100 m200 m300 m400 m500 m600 m700 m750 m790 mpath

Figure 6.28 – E1, calculated sonic velocities for varying pressure across the well depth

6. Case Studies – well F1 __________________________________________________________________________

135

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

m [kg/s]

Pwf [

bar]

9 5/8 '' 13 3/8 ''

Figure 6.16 – E2, well inflow and tubing performance

Figure 6.17 – E2, simulated well pressure and temperature profile


136

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.18 – E2, mixture density Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

void

frac

tion

Figure 6.19 – E2, void fraction Vs well depth


137

0

10

20

30

40

50

60

0.0 0.5 1.0 1.5 2.0velocities ratio (slip) vs/vw

dens

ities

ratio

Dw

/Ds

Figure 6.20 – E2, densities ratio Vs velocities ratio

1

10


dens

ities

ratio

log(

Dw

/Ds)

Figure 6.21 – E2, logarithm of densities ratio Vs logarithm of velocities ratio


138

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

acus

tic v

eloc

ity[m

/s]

Figure 6.22 – E2, acoustic velocity Vs well depth

0

200

400

600

800

1000

1200

1400

0 1 2 3 4 5 6 7 8 9 10 11time [s]

acus

tic v

eloc

ity[m

/s]

Figure 6.23 – E2, acoustic velocity Vs time


139

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

fric

tiona

l pre

ssur

e dr

op p

er 1

0m

[bar

/m] diameter change

from 13 3/8" to 9 5/8"


Figure 6.24 – E2, frictional pressure drop at 10 m distance Vs well depth

0.0

0.5

1.0

1.5

2.0

2.5

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400well depth [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

/m]

Figure 6.25 – E2, sum of frictional pressure drop Vs well depth


140

0.00

0.50

1.00

1.50

2.00

2.50

0 1 2 3 4 5 6 7 8 9 10 11time [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

]

Figure 6.26 – E2, sum of frictional pressure drop Vs time

31.5

32.0

32.5

33.0

33.5

34.0

34.5

35.0

35.5

0 1 2 3 4 5 6 7 8 9 10 11 12time [m]

Pre

ssur

e [b

ar]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0


line packing

time diferential

bubble point

Figure 6.27 – E2, estimated water – hammer and line packing at the wellhead



141

110

150

190

230

270

310

350

390

430

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

0 m100 m200 m300 m400 m500 m550 mpath

Figure 6.27 – E2, calculated sonic velocities for varying pressure across the well depth


142

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140 160 180 200 220 240 260

m [kg/s]

Pwf [

bar]

13 3/8 '' 9 5/8 ''

Figure 6.16 – F1, well inflow and tubing performance

Figure 6.17 – F1, simulated well pressure and temperature profile


143

0

100

200

300

400

500

600

700

800

900

0 200 400 600 800 1000 1200 1400 1600

well depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.18 – F1, mixture density Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 200 400 600 800 1000 1200 1400 1600

well depth [m]

void

frac

tion

[kg/

m3]

Figure 6.19 – F1, void fraction Vs well depth


144

0

20

40

60

80

100

120


dens

ities

ratio

Dw

/Ds

Figure 6.20 – F1, densities ratio Vs velocities ratio

1

10


dens

ities

ratio

log(

Dw

/Ds)

Transition between bubble and slug flow and discontinuity at the flow patterns boundary

Figure 6.21 – F1, logarithm of densities ratio Vs logarithm of velocities ratio


145

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400 1600well depth [m]

acus

tic v

eloc

ity[m

/s]

Figure 6.22 – F1, acoustic velocity Vs well depth

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20time [s]

acus

tic v

eloc

ity[m

/s]

Figure 6.23 – F1, acoustic velocity Vs time


146

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

fric

tiona

l pre

ssur

e dr

op p

er 1

0m

[bar

/m]


Diameter change from 13 3/8'' for 9 5/8''

Figure 6.24 – F1, frictional pressure drop at 10 m distance Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 200 400 600 800 1000 1200 1400 1600well depth [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

/m]

Figure 6.25 – F1, sum of frictional pressure drop Vs well depth


147

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17time [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

]

Figure 6.26 – F1, sum of frictional pressure drop Vs time

14.8

15.0

15.2

15.4

15.6

15.8

16.0

16.2

16.4

16.6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19time [m]

Pres

sure

[bar

]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4


line packing

time differential

diameter change

bubble point

Figure 6.27 – F1, estimated water – hammer and line packing at the wellhead



148

80

100

120

140

160

180

200

220

240

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

0 m100 m200 m300 m400 m500 m600 m700 m800 m900 m960 mpath

Figure 6.28 – F1, calculated sonic velocities for varying pressure across the well depth

Calculated sonic velocities

110

115

120

125

130

135

140

0.5 0.6 0.7 0.8 0.9 1

soni

c ve

loci

ty [m

/s]

0 m100 m200 m300 m400 m500 m600 m700 m800 mpath

void fraction

Figure 6.29 – F1, calculated sonic velocities for varying pressure across the well depth - scaled up.

The same plot scaled up. Acoustic velocity decreases although pressure increases across wellbore. It is caused by void fraction decrease which more affects acoustic velocity value than pressure for right-hand void values above α = 0


149

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140 160 180 200 220 240 260

m [kg/s]

Pwf [

bar]

13 3/8 '' 9 5/8 ''

Figure 6.16 – F2, well inflow and tubing performance

Figure 6.16 – F2, simulated well pressure and temperature profile


150

0

100

200

300

400

500

600

700

800

900

0 200 400 600 800 1000 1200 1400 1600well depth [m]

mix

ture

den

sity

[kg/

m3]

Figure 6.17 – F2, mixture density Vs well depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 200 400 600 800 1000 1200 1400 1600well depth [m]

void

frac

tion

Figure 6.18 – F2, void fraction Vs well depth


151

0

20

40

60

80

100

120

0.0 0.5 1.0 1.5 2.0velocities ratio (slip) vs/vw

dens

ities

ratio

Dw

/Ds

Flow patern changes from liquid only to bubble f low and then to slug f low , at the diameter change f low comes back to bubbly f low and then transforms into slug again

Figure 6.19 – F2, densities ratio Vs velocities ratio

1

10

0.0 0.0 0.1 1.0velocities ratio (slip) log(vs/vw)

dens

ities

rat

io lo

g(D

w/D

s)

Figure 6.20 – F2, logarithm of densities ratio Vs logarithm of velocities ratio


152

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400 1600well depth [m]

acus

tic v

eloc

ity[m

/s]

Figure 6.21 – F2, acoustic velocity Vs well depth

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20time [s]

acus

tic v

eloc

ity[m

/s]

Figure 6.22 – F2, acoustic velocity Vs time


153

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0 200 400 600 800 1000 1200 1400 1600 1800 2000well depth [m]

fric

tiona

l pre

ssur

e dr

op p

er 1

0m

[bar

/m] f irst bubble

appears

Diameter change from 13 3/6'' for 9 5/8''

Figure 6.23 – F2, frictional pressure drop at 10 m distance Vs well depth

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 200 400 600 800 1000 1200 1400 1600well depth [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

/m]

Figure 6.24 – F2, sum of frictional pressure drop Vs well


154

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16time [m]

Sum

fric

tiona

l pre

ssur

e dr

op

[bar

]

Figure 6.25 – F2, sum of frictional pressure drop Vs time

14.5

15.0

15.5

16.0

16.5

17.0

17.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17time [m]

Pres

sure

[bar

]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6tim

e di

ffer

entia

lline packing

time differential

diameter change

single phase

Figure 6.26 – F2, estimated water – hammer and line packing at the wellhead



155

80

100

120

140

160

180

200

220

240

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

0 m100 m200 m300 m400 m500 m600 m700 m800 m830 mpath


110

115

120

125

130

135

140

0.5 0.6 0.7 0.8 0.9 1void fraction

soni

c ve

loci

ty [m

/s]

0 m100 m200 m300 m400 m500 m600 m700 m800 m830 mpath


- scaled up. The same plot scaled up. Acoustic velocity decreases although pressure increases across wellbore. It is caused by void fraction decrease which more affects acoustic velocity value than pressure for right-hand void values

above 0.7.

Chapter 7

Discussion

7. Discussion __________________________________________________________________________

157

This chapter discuss the results of the calculations performed in present work and shown in

the form of plots. The theoretical explanation is given to the all effect observed on estimated

line packing and conclusions are linked to the field measurements. The discussion includes

the theoretical models used in calculations and verifies their limitations. The results obtained

for both oil and geothermal wells are compared to each other in order to find whether there

are other factors that may influence water-hammer and line packing in geothermal wells.

7.1 Multiphase flow correlations

In all well examples taken in considerations in present work both oil and geothermal, fluid

enters the wellbore as single liquid phase. As the average pressure and temperature existing

in tubing decrease towards the wellhead a second phase emerges due to gas coming out of the

solution in the oil wells or steam flashing in geothermal wells at the bubble point pressure.

For single phase flow physical scale and fluid property effects can be easy investigated using

mathematical modelling. For two-phase flow this is not the case. Because of the increased

number of flow conditions and flow parameters can only be obtained using wholly empirical

and semi-empirical correlations and great care must be exercised when considering the

results.

In present work two correlations was used in order to predict the pressure gradient and flow

patterns in simulated wells. The Duns and Ros correlation for oil wells and the Orkiszewski

correlation for geothermal wells. These methods were recommended in literature to give the

good pressure estimation in vertical upward flow for two-phase oil and geothermal well (Brill

and Mukherjee, 1999) and (Uphadhay and Hartz, 1977) respectively.

The Duns and Ros correlation used in PipeSim performs well and gives the results without

discontinuities. The Orkiszewski method used in HOLA appeared to have a convergence

problem in computing algorithm which results with discontinuities at the flow pattern

transitions from bubble flow to slug flow. A sudden peak observed on the plots made for

geothermal wells may be considered as the effect of these discontinuities. Some authors

7. Discussion __________________________________________________________________________

158

suggest that coefficients in equations (B.41) and (B.45) may be modified so that the slopes of

the curves are retained but discontinuities are eliminated. This solves the convergence

problem but also may affect the accuracy of results (Brill and Mukherjee, 1999).

7.2 Acoustic velocity profile

As shown in Chapter 5 the density, compressibility and acoustic velocity are the fluid

properties closely related with each other. As the properties changes across the wellbore with

respect to the pressure and temperature changes it should be expected that acoustic velocity

will be different from point to point in vertical well. The computed profile of acoustic

velocity across the wellbores depth confirms this statement.

In order to predict the sonic velocity calculations was made using formula developed by

Gudmundsson and Dong (1993) and Wood’s (1944) for oil and geothermal two-phase wells

respectively. Both formulas assume the homogeneity of the mixture and were reported to

give a good estimation with measurements. The acoustic velocity in steam-water system is

more complicated due to temperature effects and mass transfer between the phases. This

brings theoretical limitations towards the Wood’s formula, however experiments of Semenov

and Costern (1964) showed velocity in steam-water mixture agreeing with those estimated

from Wood’s, and other sound speed models have also many limitations. Gudmundsson and

Dong presented their method to be in very good agreement with measurements for gas-oil

two-phase mixture (Gudmundsson and Dong, 1993).

Speed of sound varies significantly across the two-phase length of the wellbore. In

simulations these disparities are from 100 (m/s) at the wellhead to 1150 (m/s) at the bottom

for oil wells, and likewise from 120 (m/s) to 1350 (m/s) for simulated geothermal wells.

These differences are mainly due to different densities of oil, natural gas, water and steam.

Also the fact that pressures are in different range for each of the wells is important as

analysing the estimated acoustic velocities.

7. Discussion __________________________________________________________________________

159

There are more discrepancies between acoustic velocity profiles estimated for oil and

geothermal wells. The plots for oil wells show that there is not significant drop in sonic

velocity as the crude enters the two-phase region. For geothermal wells these decrease in

sound speed is less monotonically from pure water to the points of increasing void fraction of

steam at lower pressure. Moreover in the case of well F1 and F2 where pressure at the

wellhead is of relative lower value compared to other simulated wells, acoustic velocity

decrease despite the pressure increase downwards the wellbore.

In order to explain this behaviour this was plotted on the picture were the acoustic velocity

estimated for varying pressures across the well profile was placed versus void fraction. These

plots made for gas-oil and for geothermal wells compared each other reveal different

behaviour at high void fraction values. For hydrocarbons two-phase mixture the acoustic

velocity curves have less slope which becomes more flat as the pressure is higher. For steam-

water mixture this shape is getting to be more inclined towards vertical. This more steep

shape affects the sonic velocity. In the high void fraction region in the well, where steam

occupies the main volume of the pipe, changes of void fraction value affect more acoustic

velocity that pressure changes. This makes that acoustic velocity is decreasing downwards

the well profile despite pressure increase. This effect is presented by the path on the

mentioned plots. The acoustic velocity decreases until reach the region of the lower values of

the void fraction, where the curves on the plot are more flattened. It is α = 0.751 for well F1,

and α = 0.749 for F2.

7.3 Line packing

Line packing gives the information about pressure pulse propagation in the well. In order to

estimate the line packing, frictional pressure drop across the wellbore needs to be calculated.

The sum of the frictional pressure drop versus time gives the effect of line packing after valve

closure. PipeSim gives the values of pressure drop for the every 50m distance in the case of

oil wells. For geothermal wells this was calculated with 10m step, from Darcy-Weisbach

equitation. It is obvious that this model will not give reasonable results over all flow patterns

but if the actual flow is nearly homogenous, which is the case at high volume flowrates, then

7. Discussion __________________________________________________________________________

160

predictions may be considered reasonable. Cornish (1976) applied the homogenous model to

10 different vertical oil wells and found that the percentage difference between calculated and

measured total frictional pressure drop was on average around 2%.

The time o the plots is the time it takes for pressure wave to travel from the wellhead to the

point in question and back again i.e. up and down. This pressure wave travels at the speed of

sound which is significantly lower in two phase region where also depends on void fraction.

It makes the line packing unlinearized in the two phase region. These none linearly behaviour

of line packing is clearer for low void fractions because of higher changes in sonic velocity (I

refer again to the pots of calculated sonic velocities for varying pressure across the well

depth).

The frictional pressure drop is significantly higher for two-phase flow and increase as more

gas phase arises. The plots also reveal the dependence the frictional pressure drop on

diameter size. Diameter reduction causes the increase of frictional pressure drop. Both above

effect may be observed on the plots with estimated line packing. For oil wells there is no

tubing change thus only bubble point response affect calculated line packing. For geothermal

wells two stage tubing was assumed in calculations and in this case response on both effect is

visible. These effects are most distinguishable on the plots performed for well F1. The

different line inclination may be observed. The first indicates the diameter change from 13⅜

to 9⅝” and the second is the response of bubble point. These effects are obviously visible on

the all plots made in this work, but as calculations revealed bubble point in simulated

geothermal wells tends to appear near the diameter change, and thus both effects overlap each

other what makes them less distinguishable. The time differential of the pressure dp/dt placed

together with line packing on the plots allows observing this effect.

The transitions observed on the line pacing may be important for flow condition analysis in

geothermal wells, to estimate the depth at which scale precipitation would commence for

various wellbore diameters and mass flowrates. This can assist the engineer in the selection

of operating conditions that will tend to cause scaling at shallower depth, thus required easer

7. Discussion __________________________________________________________________________

161

clean-up operations (Ragnarsson, 2000). In the case of oil wells flow condition analysis may

be used in order to optimize gas lift operations (Gudmundsson at al, 2001).

7.4 Size of the pressure pulse

Closure of a valve will create a pressure pulse. The water-hammer theory estimate the

magnitude of the pressure pulse as: the product of the sonic velocity, the density, and flow

velocity. In producing gas-liquid wells the pressure increase with depth and hence the

mixture density and speed of sound also will increase. The hypothetical valve placed at an

increasing up-stream distance will experience an increasing water-hammer.

Theoretical water-hammer at the wellhead was examined from the calculations. The results

are placed in the tables. The acceleration pressure (water-hammer) values calculated for oil-

gas two-phase wells are in range from 1.9 to 5.2 (bar) depending on the wellhead conditions.

For geothermal well these values are significantly lower and are in range of 0.6 to 0.9 (bar),

This is mainly due to wide tubing diameter that effect in decrease of fluid velocity which

directly influence water-hammer.

Chapter 8

Conclusions

8. Conclusions __________________________________________________________________________

163

1. Water-hammer and line packing were estimated for oil and geothermal wells from

calculations performed in present work. PipeSim 2001 and HOLA 3.1 multiphase flow

simulators were used to predict two phase flow in the oil and geothermal wellbores

respectively. The simulations in present work incorporate correlations developed by Duns

and Ros for oil wells and Orkiszewski for geothermal wells, coupled with equations for phase

behaviour and wellbore heat-loss. Both models assume the changes in flow regimes and

slippage between the phases.

2. The results confirmed the possibility to adapt the information contained in the rapid

pressure transients when valve is activated to determine the mixture mass flowrate, density,

velocity and gas void fraction. Additional calculations allow converting this data to the

volume flowrates of each phase. Pressure Pulse Technology may be also used for flow

condition analysis as the effect of bubble point and diameter change affect the line packing

and are readily distinguishable in both oil and geothermal cases.

3. Pressure pulse propagation is closely related to sound velocity. Speed of sound varies

significantly across the two-phase length of the wellbore. Calculations showed that velocities

are from 100 (m/s) at the wellhead to 1150 (m/s) at the bottom for oil wells, and likewise

from 120 (m/s) to 1350 (m/s) for simulated geothermal wells. These differences are mainly

due to different densities of oil, natural gas, water and steam.

Performed calculations revealed that there is not significant drop in sonic velocity as the

crude enters the two-phase region in oil wells. For geothermal wells these decrease in sound

speed is less monotonically from pure water to the points of increasing void fraction of steam

at lower pressure.

Moreover in the case of well F1 and F2 where pressure at the wellhead is of relative lower

value compared to other simulated wells, acoustic velocity decrease despite the pressure

increase downwards the wellbore. This effect is presented on the plots. Acoustic velocity

decreases until reach the region of the lower values of the void fraction, where the curves in

the plots are more flattened. It is α = 0.751 for well F1, and α = 0.749 for F2.

8. Conclusions __________________________________________________________________________

164

4. The geothermal wells studied in present work are characterized by large tubing diameters.

Typical values assumed in present work are 13 ⅜”, 9 ⅝” of producing casing, what may be

considered as very unusual compared to 6.184”, 5.125” and 4.5” taken for oil wells. This

wide diameters result with low velocities of the flowing fluid. The water-hammer theory

estimate the size of the pressure pulse as: the product of the sonic velocity, the density, and

flow velocity. This acceleration pressure (water-hammer) values calculated for oil-gas two-

phase wells are in range from 1.9 to 5.2 (bar) depending on the wellhead conditions. For

geothermal well due to large tubing size these values are significantly lower and are in range

of 0.6 to 0.9 (bar). This fact may affect the accuracy of the measurements, as the water-

hammer and line packing shape may be less distinguishable from usual pressure fluctuations

at the well head.

165

References

9. References __________________________________________________________________________

166

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Aggour M.A. Al-Yousef H.Y., Al-Muralkhi A.J. –”Vertical Multiphase Flow Correlations for

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Al-Taweel A.B, Barlow S.G – “Field Testing of Multiphase Meters”, SPE Annual Technical

Conference and Exhibition, Houston, 1999. SPE – 56583

Ansari A.M., Sylvester N.D., Brill J.P. - “Supplement to SPE 20630, A Comprehensive

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SPE Journal,1994. SPE - 28671

Bjornnson G. –“Multi-Feedzone Geothermal Wellbore Simulator”, Lawrence Berkley, report

LBL -23546, USA, 1987

Brill and Mukherjee J.P., Mukherjee H. – “Multiphase Flow in Wellbores”, SPE Monograph,

1999

Busaidi K., Bhaskaran H., – “New Development in Water Cut Meter with Salinity

Compensation”Petroleum Development Oman (PDO), Oman, J. Chen, Haimo Technologies,

China, 2002. SPE – 77894

Campbell – “Gas Conditioning and Processing” Vol.1 The basic Principles, Campbell

Petroleum Series, USA, 1994

Cornish R. E. – “The Vertical Multiphase Flow of Oil and Gas at High Rates” Abu Dhabi

Petroleum Co. Ltd., SPE Journal, 1976. SPE – 5791

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Appendix A

Multiphase Metering Projects

Appendix A – Multiphase Metering Projects __________________________________________________________________________

175

Following text contains a brief description of some MFMs projects that are now

commercially available. The tables given below the text A.1, A.2, A.3, A.4, contain

comparison of the methods with regard to the techniques that are used for measurement

purposes.

Roxar AS MFI

Several elements are combined to measure the phase flow rates. The phase fraction sensor is

mounted vertically with microwave transmitters and receivers protruding a few millimetres

into the flow passage. In the fraction sensor, a resonant cavity is contained between two sets

of microwave absorbers. Measurements of the frequency in the cavity are claimed to be

proportional to water cut in the liquid phase. A single-energy gamma densitometer is used to

measure the total mixture density. The gas fraction and the water cut are derived from this

measurement. The mixture velocity is gained from cross-correlation between two microwave

sensors located a known axial distance apart. (Stokes et al., 1998)

Roxar AS Fluenta

This meter uses several different sensors in combination. Capacitance and inductance sensors

are used to measure bulk electrical properties of the flowing mixture in oil and water

continuous flows respectively, with the water cut derived from these measurements. A single

energy gamma densitometer measures the average mixture density by attenuation of gamma

photons. The phase fractions can then be extracted from this information, as described earlier.

Velocity measurement is by cross-correlation of capacitance signals for oil continuous flow,

and by differential pressure across a venturi when the flow is water continuous. Velocity and

phase fraction measurements are then combined to give phase flow rate information. (Warren,

et al., 2001)


176

Kvearner-DUET

This flowmeter uses the attenuation of gamma rays at two different energies to derive the oil,

water and gas phase fractions. The mass absorption coefficients of oil and water vary as a

function of gamma photon energy and so the two different absorption rates and continuity

relationship allow the phase fractions to be determined. To maximise the transmission of the

lower energy gamma rays the sources and detectors are arranged around a GRP pipe section.

Velocity measurement is by cross-correlation of two gamma densitometer signals, so it

responds most accurately to distinct multiphase flow features like liquid slugs. (Roach,

Whitaker, 1999)

Schlumberger-Framo

A mixer, which it is claimed gives both spatial and temporal mixing, is utilised to pre-

condition the flow entering a venturi. The mixer consists of a large plenum chamber and

piccolo tube. The piccolo tube penetrates the base of the plenum chamber and conducts the

flow to the venturi, the aim being to draw the gas and liquid into the venturi at equal velocity.

The differential pressure across the venturi is proportional to the total volume flow rate. A

dual-energy gamma densitometer is mounted at the throat of the venturi and is used to derive

phase fractions. (Hansen, 1997)

Jiskoot Mixmeter

The Mixmeter makes use of two separate radioactive sources to measure both the phase

fractions of the multi phase mixture and the mixture velocity. An integral part of the

flowmeter is a static mixer which conditions the mixture (spatially) so that an even

distribution of the phases is maintained at the measurement cross-section. The phase fractions

are determined by taking radiation attenuation measurements over a spectrum of energies.

Passage of the gamma rays through the pipe is facilitated by the use of low absorption

windows. Determination of the mixture velocity is by cross-correlation of photo detector

signals received from two sources of equal energy, but mounted a known axial distance apart.


177

Differential pressure measurement across the static mixer is also used as secondary measure

of flow velocity. (Dowty et al., 1991)

FlowSys

The FlowSys multiphase flowmeter uses two arrays of vertically mounted capacitance

sensors, in what is effectively a very sophisticated liquid level sensing system. Two thin

parallel plates are mounted axially in the flow passage, at a known distance apart. An array of

capacitance sensors is mounted on the surface of each plate. The capacitance of the flow is

thus measured at several vertical intervals in the flow passage. The gas/liquid interface is

determined and the gas fraction calculated. The average water cut of the liquid is derived

from the capacitance between the lower sensor pairs below the gas/liquid interface. The meter

only operates in the slug flow regime, as this is a requirement for effective cross-correlation

between the two sets of capacitance sensors. A slip correlation is then used to estimate the

bulk gas and liquid velocities. The current version for oil continuous flow is being up-dated

for operation in water continuous flows. (Al-Taweel, Barlow, 1999)

Agar

The flowmeter contains a rotary positive displacement flowmeter, modified for multiphase

use, and two Venturis in series in a vertically upward flow. An algorithm in the control

computer derives the gas and liquid volume flow rates from these outputs. The water content

of the flow is derived from the power absorbed by the process fluid from an in-line

microwave monitor, and the continuous liquid phase is detected by the phase shift between

the transmitter and two differentially spaced aerials. The measurement of the liquid phase

water cut can then be derived from the gas fraction and the microwave monitor output.

Individual oil, water and gas flow rates are then computed from these parameters. (Agar, and

Farchy, 2002)


178

ISA Scrollflow

Essentially this is a positive displacement flowmeter. It has two counter-rotating shafts which

are machined to form a continuous and constant volume cavity, and the rotation created to the

shafts by the fluid passing through the meter is claimed to be proportional to the total volume

flow rate. A single-energy gamma densitometer located at the centre of the meter measures

the overall mixture density. If the water cut of the liquid phase is known, then the phase flow

rates can be determined from these measurements. At present the meter design does not

include an integral water monitor to determine liquid water fraction. (Millington, 1999)

Haimo

The Haimo multiphase flowmeter, developed and used in China, uses an electromagnetic

valve to periodically switch flow through a bypass trap where liquid collects. The velocity of

this liquid passing through the meter is measured by cross-correlation between two single-

energy gamma densitometers. The gas velocity is derived using a slip relationship. (Busaidi

et al., 2002)

Esmer

The Esmer multiphase flowmeter, currently under development, uses a neural network

approach to interpret signals from capacitance/conductance sensors and pressure transmitters,

combining measurements from 'training' of the meter to predict the expected values of phase

velocities. (Toral, at al., 1999)

Tables below contain comparison of the methods with regard to the techniques that are used

for measurement purposes.


179

Table A.1 – Methods utilizing Flow Conditioning

Flow Conditioning

Homogenisation Leave-as-it-is In-line separation

Jiskoot-Mixmeter

Schlumberger-

Framo

TEA-Lyra

ISA- Scrollflow

ISA-Solarton

Schlumberger-VX

Roxar AS Fluenta

Roxar AS-MFI

Kvearner-DUET

Megra (for GVF < 25%)

Esmer

Yokogawa

Agar

WellCamp

Accuflow

Kvearner-CCM

Megra (for GVF > 25%)

Haimo

Jiskoot-Starcut


180

Table A.2 – Methods utilizing a Gamma Source

Gamma Source

Used Not Used

Roxar AS Fluenta

Schlumberger-Framo

Schlumberger-VX

TEA-Lyra

ISA- Scrollflow

Kvearner-DUET

Megra

Jiscoot-Mixmeter

Agar

Esmer

WellComp

Jiskoot-Starcut

Kvearner-CCM

TEA-Vega

ISA-Solarton

TEA-Lyra (for WC < 25%)

FowSys

Yokogawa


181

Table A.3 Methods utilizing Intrusive methods

Intrusive

Yes No(*)

Jiscoot-Mixmeter

Agar

Schlumberger-Framo

Accuflow

WellComp

Kvearner-CCM

ISA- Scrollflow

ISA-Solarton

TEA-Vega

Jiskoot-Starcut

Haimo

(*)Venturi’s are not regarded as intrusive devices

Schlumberger-VX

Roxarn AS-Fluenta

ROxar AS-MFI

Kvearner-DUET

Esmer

Megra

TEA-Lyra

FlowSys

Yokogawa


182

Table A.4 Methods utilizing Cross Correlation

Cross Correlation

Yes None

Roxar AS-MFI

Kvearner-DUET

Roxar AS-Fluenta

FlowSys

Yokogawa

Haimo

Jiskoot-Mixmeter

Agar

Accuflow

WellComp

Jiskoot-Starcut

Kvearner-CCM

ISA-Scrollflow

ISA-Solarton

Esmer

Megra

Schlumberger-VX

TEA-Lyra

Schlumberger-Framo

TEA-Vega

Appendix B

Duns and Ros, Orkiszewski - Multiphase Flow Correlations

Appendix B – Duns and Ros, Orkiszewski – Multiphase Flow Correlations __________________________________________________________________________

184

B.1 Duns and Ross correlation

Flow-Pattern Prediction

Figure B.1 shows the flow-pattern map developed by Duns and Ros. They identified four

separate regions for computation purposes, Regions I through III and a transition region.

Duns and Ros also identified the heading region as a fifth region, but this is now considered

part of Region II. In this work I will refer to Regions I through III as bubble, slug, and mist

flow, respectively.

Figure B.1 Duns and Ros – Flow Patterns Map

The flow-pattern transition boundaries are defined as functions of the dimensionless gas and

liquid velocity numbers NGu and NLu. For these transition boundaries, Duns and Ros

proposed these equations.

Bubble/slug boundary:

uNLLuN LSBG ⋅+= 21/ (B.1)


185

where L1 and L2 are functions of Nd – dimensionless diameter number. The gas and liquid

velocity numbers are given as:

4

L

LLL g

uuNσρ⋅

⋅= (B.2)

and

4

G

GGG g

uuNσρ⋅

⋅= (B.3)

Where u is the superficial velocity, g – gravity acceleration, and ρ density of the gas and

liquid indicated with subscripts.

Slug/transition boundary

uNuN LTrSG ⋅+= 3650/ (B.4)

Transition/mist boundary

75,0

/ 8475 uNuN LMTrG ⋅+= (B.5)

Liquid holdup prediction: Duns and Ros chose to develop empirical correlations for a

dimensionless slip-velocity number - NKu, rather than for liquid holdup. NKu is defined in

similar way to the gas and liquid velocity numbers.

4

L

LKK g

uuNσρ⋅

⋅= (B.6)

The slip velocity was defined as

L

LK

L

GKLGK H

uH

uuuu −

−=−=

1 (B.7)


186

where HL is the liquid holdup.

Pressure gradient contains three components: gravitational, frictional, and accelerational.

accK

f dzdpg

dzdp

dzdp

⎟⎠⎞

⎜⎝⎛+⋅+⎟

⎠⎞

⎜⎝⎛= ρ (B.8)

In order to calculate the gravitational pressure gradient following procedure was proposed

- Calculate the dimensionless slip velocity NKu, using the appropriate correlation.

The correlations for NKu are different for each flow pattern and are given later.

- Solve the equations given above for slip velocity.

- Calculate the liquid holdup from equation given above.

- Calculate the slip density from equation (B.9)

( )LGLLK HH −⋅−⋅= 1ρρρ (B.9)

- Calculate the elevation component of the pressure gradient as given in equation (B.8)

Flow pattern identification:

- Bubble flow exists if

SBGG uNuN /< (B.10)

For Bubble flow, the dimensionless slip-velocity number is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅+⋅+=uN

uNFuNFFuN

L

GLK 1

'321 (B.11)


187

where F1 and F2 are function of the liquid velocity number NL. F3’ can be obtained from

dNF

FF 433 ' ⋅= (B.12)

where F4 and Fd are also the functions of NL.

The friction pressure-gradient component for bubble flow is given by

duuf

dzdp MLL

f ⋅⋅⋅⋅

=⎟⎠⎞

⎜⎝⎛

2ρ (B.13)

From experimental data Duns and Ros developed the equation for friction factor:

3

21 f

fff ⋅= (B.14)

The friction factor is governed mainly by f1, which is obtained from a Moody diagram Fig.

(B.2) as function of Reynolds number for the liquid phase.

L

LL duµ

ρ ⋅⋅=Re (B.15)

The factor f2, is a correction for the in-situ gas/liquid ratio. The factor f3 is considered by

Duns and Ros as second-order correction factor for both liquid viscosity and in-situ gas/liquid

ratio. It becomes important for kinematics viscosities greater than approximately 50 cSt

(0,744 Pa·s) and is given by

L

G

uuf

f⋅

⋅+=504

1 13 (B.16)

Duns and Ros considered the acceleration component of the pressure gradient to be neglected

for bubble flow.


188

- Slug flow exist if

TrSGGSBG uNuNuN // << (B.17)

For slug flow the dimensionless slip-velocity umber is

( ) ( )uNFFuN

FuNL

GK ⋅+

+⋅−=

7

6982,0

5 1'

1 (B.18)

where F5, F6 and F7 are function of the liquid viscosity number NµL, and

66 029,0' FNF d +⋅= (B.19)

Figure B.2 Moody diagram


189

The friction pressure gradient component for slug flow is calculated exactly the same way as

the bubble flow. Also, the accelerational component slug flow is considered negligible.

Mist flow exist if

MTrGG uNuN /> (B.20)

Duns and Ros assumed that, at high flow rates, the liquid is transported mainly as small

droplets. The result is nearly a no slip condition between the phases. Thus NKu = 0 and uK = 0

and HL = (1-α). The mixture density for use in elevation component of the pressure gradient

then is calculated from

( ) LGM ραραρ ⋅−+⋅= 1 (B.21)

Friction in the mist flow pattern originates from the shear stress between the gas and the pipe

wall. Thus, the friction component of the pressure gradient is determined from

duf

dzdp GG

f ⋅⋅⋅

=⎟⎠⎞

⎜⎝⎛

2

2ρ (B.22)

Because there is no slip, the friction faction is obtained from a Moody diagram Figure (B.2)

as a function of a Reynolds number for gas phase

G

GS duµ

ρ ⋅⋅=Re (B.23)

Duns and Ros noted that the wall roughness for the mist flow is the thickness of the liquid

film that covers the pipe wall. Waves on the film cause an increased shear stress between the

gas and the film that, in turn, can cause the greatest part of the pressure gradient. These

waves result from the drag of the gas deforming the film in opposition to the surface tension.

This process is affected by viscosity and also is governed by a form of the Weber number


190

L

GGWe

uN

σερ ⋅⋅

=2

(B.24)

This influence was accounted for by making a new function of dimensionless number

containing liquid viscosity.

εσρµ

µ ⋅⋅=

LL

LN2

(B.25)

The value of roughness may be very small, but the relative roughness never becomes smaller

than the value of the pipe itself. At the transition to slug flow, the waviness may become

large, with the crests of opposite waves touching and forming liquid bridges. Then ε/d

approaches 0.5. Between these limits, ε/d can be obtained from equations

dudN

GG

LWe ⋅⋅

⋅=≤ 2

0749,0;005,0ρ

σε (B.26)

and

( ) 302,02

0371,0;005,0 µρσε NN

dudN We

GG

LWe ⋅⋅

⋅⋅⋅

=> (B.27)

where d is in [ft] unit, and uG is in [ft/s] ρG in pounds per cubic foot and σ in dynes per

centimetre. Values of f for the mist flow-pattern can be found for ε/d > 0,005 from

extrapolation of the Moody equation.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛⋅+

⎟⎠⎞

⎜⎝⎛ ⋅⋅

⋅=73,1

067,027,0log4

14d

d

f εε

(B.28)

As the wave height of the film on the pipe wall increases, the actual area variable for gas

decreases because the diameter open to flow of gas in now d-ε. Duns and Ros suggested that


191

the friction component of the pressure gradient could e refined by replacing d with d-ε and uG

with uG · d2/( d-ε)2 throughout the calculations. This results a trial-and error procedure to

determine the ε. In mist flow, acceleration often can not be neglected as it was in bubble and

slug flow. The accelerational component of the pressure gradient cab be approximated by

⎟⎠⎞

⎜⎝⎛⋅⋅

=⎟⎠⎞

⎜⎝⎛

dzdp

puu

dzdp MGM

acc

ρ (B.29)

The derivation f this equation provided by Beggs and Brill and Mukherjeecan be found in

Brill and Makherjee monograph (1999).

- Transition region exists if

MTrGGTrSG uNuNuN // << (B.30)

If this region is predicted, Duns and Ross suggested linear interpolation between the flow-

pattern boundaries, to obtain the pressure gradient. This will require a calculation of pressure

gradient with both slug-flow and mist-flow correlations. The pressure gradient in the

transition region then is calculated from

( )mistslugf dz

dpAdzdpA

dzdp

⎟⎠⎞

⎜⎝⎛⋅−+⎟

⎠⎞

⎜⎝⎛⋅=⎟

⎠⎞

⎜⎝⎛ 1 (B.31)

Increased accuracy was claimed in the transition region if the gas density used in the mist-

flow pressure-gradient calculations was modified to be

MTrG

GGG uN

uN

/

' ⋅= ρρ (B.32)

where ρG is the gas density calculated at the given conditions of pressure and temperature.

This modification accounts for some of the liquid being entrained in the gas.


192

B.2 Orkiszewski correlation

- Bubble/slug transition:

BSB L=/α (B.33)

du

L MB

2

2218,0071,1 ⋅−= (B.34)

where uM is in [ft/s] and d in [ft]. LB is constrained algebraically to be ≥ 0.13.

- Bubble flow exist if

BL≤α (B.35)

The liquid holdup for bubble flow is determined from

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⋅−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−+⋅−=

K

G

K

M

K

ML u

uuu

uu

H 411211

2

(B.36)

which is equivalent to equation for Duns and Ros correlation. Orkiszewski adopted the

Griffith suggestion that 0.8 [ft/s] is a good approximation of the average uK that is function of

the gas liquid densities and surface tensions (Brill, 1999). The liquid holdup determined from

above equation is then used to calculate slip density with equation (B.9) which in turn is used

to calculate the elevation component of pressure gradient.

The friction pressure-gradient component for bubble flow is given by

dHu

f

dzdp L

LL

f ⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

=⎟⎠⎞

⎜⎝⎛

2

2

ρ (B.37)


193

The friction factor is obtained from a Moody diagram Figure (B.4) as function of relative

roughness and Reynolds number for the liquid phase

L

L

LL

L

dHu

µ

ρ ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

=

2

Re (B.38)

The acceleration pressure-gradient component for bubble flow was considered negligible.

- Slug flow

Slug flow exist if inequalities are satisfied

SB /αα > (B.39)

and

TrSGG uNuN /< (B.40)

The slip density is calculated from

( )Γ⋅+

+⋅++⋅

=L

BM

GGBLLK uu

uuuρ

ρρρ (B.41)

Orkiszewski developed above equation (B.41) by performing mass and volume balances on a

typical slug unit consisting of a bubble and liquid slug. A similar Griffith and Wallis

development neglected the presence of a liquid film around the bubbles ad the possibility of

liquid droplets being entrained in the bubbles. Consequently Orkiszewski used the data of

Hagedorn and Brown and proposed the last term in equations (B.41), to account for the

distribution of liquid in this region. This modification was meant to extend the Griffith and

Wallis work to include the high-velocity flow range. Griffith and Wallis correlated the

bubble-rise velocity uB by the relationship

dgCCuB ⋅⋅⋅= 21 (B.42)


194

where C1 and C2 are the functions of ReB and ReL. The precise procedure of uB calculations

including iteration process for higher Reynolds number may be found in Brill and

Mukherjeeand Makherjee monograph (1999).

L

BLB

duµ

ρ ⋅⋅=Re (B.43)

and

L

MLL

duµ

ρ ⋅⋅=Re (B.44)

The friction pressure-gradient is given by

Γ+⎟⎟⎠

⎞⎜⎜⎝

⎛++

⋅⋅⋅⋅⋅

=⎟⎠⎞

⎜⎝⎛

BM

BLML

f uuuu

dguf

dzdp

2

2ρ (B.45)

The friction factor f is calculated from the Moody diagram using the following definition of

Reynolds number

L

MLB

duµρ ⋅⋅⋅

=1444

Re (B.46)

Pressure drop due to acceleration is neglected in slug-flow pattern.

For transition and mist flow regime Orkiszewski uses the Begs and Brill methods that are

described above in previous section of the Appendix B.

Appendix C

Sound Wave Propagation Process in Steam Water Mixture

Appendix C – Sound Wave Propagation Process in Steam Water Mixture __________________________________________________________________________

196

The complex physical process that occurs during propagation of sound wave in water-vapor

two-phase system was described by (Kieffer, 1977).

Consider the temperature-entropy diagram of water shown above on Figure (5.7). A mixture

of saturated water and steam is represented as point G where the chord ratio FG/FH is the

mass fraction x of steam and the mixture. Isentropic pressure changes, such as the

compression and rarefaction which occurs during propagation of sound wave are presented

by movement up and down the constant entropy line CGK. If steam and water remain in

thermal equilibrium on the saturation line, there must be mass transfer between the phases,

since the fraction of steam in mixture changes (BC/BD ≠ FG/FH ≠ IK/IM). This requires that

condensation or evaporation take place. An isentropic pressure increase from p to p + ∆p

corresponds to movement of the mixture from G to C in the temperature-entropy diagram.

The pressure increase in the water phase corresponds to movement from F to A; as a result,

the water phase becomes subcooled. The pressure increase in the steam phase corresponds to

movement from H to E as the steam becomes superheated. The induced temperature

difference between the steam and the water leads to heat transfer from the superheat steam to

subcooled water. If the original composition of the mixture G lies to the right of the peak of

the two-phase loop as shown in (Figure 5.7), some water will be vaporized and the mass

fraction of steam in the mixture will increase during adiabatic compression (BC/BD >

FG/FH). If the original composition lies to the left of the top of the two-phase loop (assumed

to be symmetric), some steam will condense, and the mass fraction of steam in the mixture

will decrease during adiabatic compression. Thus by heat and mass transfer both water and

the steam are restored to the saturation line, the water by the path A to B, and the steam by the

path E to D. An isentropic pressure reduction from p to p - ∆p corresponds to the movement

from G to K in the temperature - entropy diagram. The pressure decrease in the water phase

corresponds to movement from F to R; as result, the water becomes superheated above its

saturation temperature point I (IR is a continuation of the pressure line from the water region).

The pressure decrease in the steam phase alone corresponds to movement from H to N on the

diagram, and the steam becomes subcooled or supersaturated with respect to its saturation

temperature, point M (MN is a continuation of a constant pressure line from the superheated

region). If the original composition of the mixture G lies on the right side of the two-phase

Appendix C – Sound Wave Propagation Process in Steam Water Mixture __________________________________________________________________________

197

loop, some vapor will condense to form a mixture of saturated water and steam (point L), and

thus the subcooled steam may move toward the stable state M. The mass fraction of steam in

such a mixture will decrease (IK/IM < FG/FH). If the original composition lies to the left of

the two-phase loop (assume to be symmetric), cavitation, and some vaporization of the

subcooled liquid will be dominant process and the mass transfer of steam in the mixture will

increase. Cavitation of the water creates a local mixture with the composition of point J and

the superheated water at point R thus moves towards the stable state I, a mixture of water and

steam.

Figure C.1 Temperature – entropy diagram for H2O

Appendix D

PipeSim 2000-Multiphase Flow Simulator

Appendix D – PipeSim 2000 - Multiphase Flow Simulator __________________________________________________________________________

199

D.1 PipeSim Well Performance Analyses

PipeSim is the program developed by Baker Jardine & Associates, London. The program

allows predicting the relationship between flowrates, pressure drop, and piping geometry

(length, diameter, angle, etc.) for the fluids produced from reservoir. Individual PipeSim

modules are used for a wide range of analyses including: well monitoring, nodal analysis,

artificial lift optimalisation, pipeline and process facilities modelling, and field planning. A

major future of PipeSim 2000 is the system integration and openness that allows constructing

the total production model from reservoir to processing facilities. The model can be either

operated hooked up to the reservoir simulator, such like Eclipse, or with a simplified material

balance. The simulations can track multiphase issues such as slugging, fluid velocities, or

thermal performance and also be used to monitor topside equipment such as compressor

power requirements. Program allows deciding between the customary and SI units and

enables the different input and output units depending on which are more convenient for

calculations.

D.1.1. Fluid Properties Correlations

At the beginning the program requires the fluids properties to be specified. To predict

pressure and temperature changes from the reservoir along the wellbore (or flow line tubular),

it is necessary to predict fluid properties as a function of pressure and temperature.

PipeSim has a PVT section which can generate fluid properties using standard correlations

and allows them to be modified. In present work The Black Oil Glasø correlation has be used

to calculate the fluid properties. The Black oil correlations futures are described in chapter 5

of this work which deals with Multiphase flow in wells. Glasø PVT correlation has been

recommended for light North Sea crude by PipeSim Manual authors (PipeSim Manual)

developed from analysis of crude from the following fields: Ekofisk, Statfiord, Forties,

Valhall.


200

Figure D.1 Black Oil PipeSim input data window

The mathematical relationship between bubble point and solution gas oil ratio Rs given by

Glasø is 2255.1

172.0

989.0

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅= B

APIGs p

TR

γγ (D.1)

where T is in oF, 5.1315.141−=

OAPI γ

γ , γG is the specific gravity of gas (air = 1), γO is the

specific gravity of oil

The bubble point pressure is given by the relation

( )( )( )5.0log3093.31811.148869.210 p

Bp ⋅−−⋅= (D.2)

The other models available for Black Oil correlations in program are: Lasater, Standing,

Vasques and Begg. The Standing correlations are widely used in oil industry. They are based

primarily on California crude oils and these correlations do not correct for other oil types or


201

hydrocarbon content. Glasø modified the Standing correlation to make the independent of oil

type (Glasø, 1980).

Glasø’s oil formation volume factor correlation can be expressed mathematically as

A

oB 100.1 += (D.3)

where

( )2*log27683.0*log91329.258511.6 obob BBA −⋅+−= (D.4)

and B*ob is correlating number defined by

TRBO

Gsob ⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅= 968.0*

526.0

γγ

(D.5)

Glasø also presented the bubble point pressure. The proposed bubble point pressure

correlation can be expressed as (Glaso, 1980)

( )2*log30218.0*log7447.17669.1log bbb ppp ⋅−⋅+= (D.6)

with p* b a correlating number defined by

989.0

172.0816.0

*APIG

sb

TRp

γγ⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛= (D.7)

where Rs is in scf/STBO, T in oF, and γG is the total specific gravity of the surface gases.

Optionally the program allows defining the viscosity data for the required pressure and

temperature ranges. The viscosity of crude oil with dissolved gas is an important parameter in

pressure-loss calculations for flow in pipes. There are four steps to calculating the liquid

viscosity as follows:


202

Step 1: Calculate the dead oil viscosity at atmospheric pressure and the flowing fluid

temperature.

Step 2: Calculate the saturated live oil viscosity at the flowing fluid pressure and temperature

assuming that the oil is saturated with dissolved gas.

Step 3: Establish if the flowing pressure is above the bubble point pressure for the flowing

fluid temperature. If not, continue to step 4, otherwise calculate the undersaturated oil

viscosity.

Step 4: Determine the viscosity effects of water in the liquid phase.

Figure D.2 Viscosity data input window

The dead oil viscosity can be also calculated from Glasø Correlation. He presented an

empirical correlation based on North Sea data (Glaso, 1980). The correlation can be

expressed as

( ) ( ) 447.36log10141.3 log313.10444.310 −⋅⋅⋅= ⋅− TAPIod T γµ (D.8)

where T is in oF.


203

Saturated live oil viscosity depends on the solution-gas content. Oil viscosity decreases with

rising pressure as the solution gas increases, up to the bubble point pressure. There are few

correlations to determine the viscosity of saturated oil system, the Begs and Robinson

correlation has been chosen in present work. The empirical form of this equation is (Brill,

1999)

( )[ ] bodsO R µµ ⋅+⋅= − 515.0100715.10 (D.9)

where

( ) 338.015044.5 −+⋅= sRb (D.10)

Above the bubble point pressure, rising pressure increases the viscosity o foil because of its

compressibility. The correlation proposed by Vasquez and Beggs is the one of available in

PipeSim to calculate the undersaturated oil viscosity.

a

bObO p

p⎟⎟⎠

⎞⎜⎜⎝

⎛⋅= µµ (D.11)

where

b

pa 10187.16.2 ⋅⋅= (D.12)

and

( ) 55109.3 −− ⋅⋅−= pb (D.13)

D.1.2 Advanced calibration data

In many cases, actual measured values for some properties show a slight variance when

compared with the value calculated by the black oil model. In this situation it is useful to

"calibrate" the property using the measured point. PipeSim can use the known data for the

property to calculate a "calibration constant" Kc .


204

),(

),(

TPpropertycalculated

TPpropertymeasuredK C =

This calibration constant is then used to modify all subsequent calculations of the properties.

Properties which may be calibrated in this manner are; oil formation volume factor, gas

compressibility, live oil viscosity, gas viscosity.

Additionally PipeSim allows entering the fluid composition data

D.2 Profile model

I these work the vertical well performance analysis model has been selected for simulations.

The other models available in the program are; injection well performance, and surface and

facilities model. The program requests only the well hardware data required by the option

which was selected. This future is convenient because for particular section calculations full

model do not have to be built up, and I present work the surface equipment could be skipped.

The main window displays the designed model Figure D.3 and allows for easy access to each

of the section in order to model the required parameters.

Figure D.3 Designed model


205

The profile model section of the program allows describing the production tubing, SSSV

(Surface - Controlled Subsurface Safety Valve, Figure D.4), and restriction. Downhole

equipment parameters may be specified in program using one of the two options available,

the simple and detailed model. The displayed screed is different depending on the option

selected. Using the detailed model more factors to be taken into calculations but obviously

requires more data to be entered.

Figure D.4 Surface - Controlled Subsurface Safety Valve

D.2.1. Detailed model

The detailed profile contains following sections:

- Deviation survey

This section has been left empty in present work as the vertical well model has been assumed.

If such deviation data are available it may be copied to the program from other spreadsheet

applications.


206

- Geothermal survey

The geothermal survey and overall heat transfer coefficient need to be specified in the

detailed model. The program request at least two values at the wellhead and at the bottom.

The heat transfer coefficient describes the resistance to heat flow by all mechanisms

(convection, radiation and conduction) from the well to its surroundings. The program

proposes the typical default values of the coefficient that maybe used in calculations.

Temperature prediction may be performed using “enthalpy balance” temperature model. This

requires to define the well environment including all casing, string, cement tops, formation

lithology etc.

-Tubing configuration

The data entered in this section determines extend of tubing modelled. The program takes the

bottom of the last tubing as the fluid inflow point. Program offers the tubing tables containing

standard values of tubing and casing diameters. Also the wall thickness and roughness may

be specified using values given in the PipeSim tubing tables. Casing inner diameters are only

required if the flow is set to be annular or both tubing and annulus.

-Downhole equipment

This option allows specifying the necessary data depending on the equipment was selected.

Program allows to set up the following optional equipment properties: gas lift injection points,

ESP (Electrical Submersible Pumps), Choke, separator G/L (Gas/Lift) valve system, SSSV.

Perforations data may be also entered in this mode.

D.2.2. Simplified model

For present work simple model was utilized Figure D.5. This model requires the limited

number of information compared to the detailed motel described above. Deviation survey is

replaced by the one deviation angle from horizontal, where the kick off depth may be


207

specified. Temperate data are limited to the surface ambient and reservoir (well bottom).

Tubing limited up to the four sections, and if more are required detailed profile need to be

used. The downhole equipment assumed in simple mode contains only SSSV, G/L vale and

perforation data.

The accuracy of the equipment description may by verified by making an equipment

summary. The program draws and displays the simple sketch of designed downhole or

surface equipment.

Figure D.5 Simple profile mode

Figure D.5 Simple profile mode

D.3 IPR Data

In present work the one oil well the IPR curve was also calculated using PipeSim simulator.

This requires additional data defining the reservoir inflow performance. The program allows

for thirty inflow options including the oil and gas and gas condensate reservoirs, to model the

flow of fluids from the reservoir, trough the formation, and into the well. The choice which of

them should be used depends on the available information and the type of sensitivities that


208

are supposed to be run. This is beyond the scope of this work to present them all and there is

the bulk of the literature which describes those models.

The average reservoir pressure and reservoir temperature must be entered for inflow

performance models; however both the Multi-rate Fetkovich and Multi-rate Jones models can

be used to calculate the reservoir pressure. Well skin can be either directly entered or

calculated using Locke, Macleod or Karakas and Tariq methods for mechanical – geometrical

skin, and the Cino/Martia -Bronz or Wong - Clifford method for a deviation-partial

penetration skin. The Elf - Skin Aide model is also available.

The calculated IPR can be matched to measured data and used to calculate the IPR pressures

for any rate and water cut value. Also relative permeability can be applied to all IPR model in

PipeSim, and thus the total mobility of oil, gas and water may be determined.

For calculations in present work the Darcy inflow equation has been selected above the

bubble point and the Vogel solution below the bubble point. Required input is: reservoir

permeability (total permeability at the prevailing watercut and Rs), reservoir thickness

(thickness of the producing reservoir rock), - drainage area, - wellbore radius, Dietz shape

factor (to account for the shape of the drainage area). The additional data for simulations has

been taken from Kleppe Manuscript (1990), where author gives the characteristic of the most

Norwegian North Sea fields. The other mentioned models available in the program take into

considerations the deliverability change with time, non Darcy flow, multi-rate flow,

hydraulically fractured well, dual porosity, horizontal and multi layer wells.

D.4. Matching option

The matching option allows user for data quality control and fine adjustment of model

parameters to enable well model to produce observed data. Appropriately matched model is

requested for accurately performance prediction. The matching option offers the following

calculation options:

- Correlation comparison.


209

This option allows pressure gradient plots to be generated with different correlations to be

compared with the measured gradient survey data and each other.

- VLP and IPR matching

This option enables to tune the wellbore multiphase flow correlations to fit a range of

measured downhole pressures and rates. Once the VLP is matched, the IPR can be adjusted to

match the observed rates and pressures also.

- Gradient matching

Exits correlations can be modified using non-linear regression to best fit a gradient survey.

Comparison of the fit parameters will identify which correlation requires the least adjustment

to match the measured data.

- Surface pipe matching

The PipeSim program uses actual wellhead and manifold pressures together with temperature

data points to match the surface pressure drop correlations. Separate screens allow the match

parameters to be viewed and the best match selected.

- Tubing/pipeline correlations parameters

The match parameters can be inspected, reset or entered by and using this option. This

capability may be useful for troubleshooting, or to input match parameters determined

previously.

- Correlation thresholds

This is a capability of the program that allows specifying a threshold angle for both tubing

and pipeline correlations at which the program will automatically change to another specified

correlation. This option is particularly dedicated to the vertical risers in subsea completions to

be modeled more accurately.


210

D.5 VLP correlations and applications

It is no universal rule for selecting the best flow correlations for a given application. It is

recommended that correlation comparison always be carried out (PipeSim Manual). By

inspecting the predicted flow regimes and pressure results, the user can select correlation that

best models the physical situation.

In present work for oil wells calculations the Duns and Ros correlation was selected. This

correlation is recommended by PipeSim authors (PipeSim Manual) for vertical well

calculations. The following calculations are compared below PipeSim contains also the

correlations developed by Shell EP - Technology and Application Research, Those

correlations are described t combine the best futures of the existing correlations (Zabaras,

2000), and also contains additional futures for predicting low-rate VLP’s well stability or for

viscous, volatile, ad foamy oils. These correlations could not be used as requires additional

extra license for its use.

- Duns and Ros

This correlation was precisely described in previous chapter 5. The description presents the

basis of the original Duns and Ros published method. PipeSim contains the primary

correlation that was enhanced and improved for use wit condensates. The Duns and Ros

correlation performs well in mist flow cases and may be used in high Rs oil wells (Brill,

1999). It tends to overprotect VLP in oil wells. Despite this, the minimum stable rate

indicated by the VLP curve is often a good estimate (PipeSim Manual).

- Fancher Brown

It is a no slip holdup correlation that is provided in use as a quality control. It gives the lowest

possible value of VLP since it neglect the gas-liquid slip. It should always predict a pressure

which is less than he measured value. Even if it gives good match to measured downhole

pressure Fancher Brown should not be used on the quantitative work. Measured data falling


211

into the left of the Fancher Brown on the correlation comparison plot indicates the problem

with fluid density or field pressure data (PipeSim Manual).

- Orkiszewski

This correlation often gives a good match to measured data. However its formulation

includes a discontinuity in the calculation method. The discontinuity can cause instability

during the pressure matching process, thus is not recommended for calculations (PipeSim

Manual).

- Begs and Brill

The correlation is primarily horizontal pipeline correlation (Brill, 1999), thus was not taken

into account for wellbore simulations. It generally over predicts pressure drops in vertical and

deviated wells.

- OLGA-S steady-state

OLGA-S is based in larger part on data from the SINTEF two-phase flow laboratory near

Trondheim, Norway. The test facilities were designed to operate at conditions that

approximated field conditions. OLGA-S considers four flow regimes, stratified, annular, slug

and dispersed bubble flow and uses unique minimum slip criteria to predict flow regime

transitions. This correlation is available to all members of the SINTEF syndicate, and to non-

members on payment of the appropriate fees.

- Govier and Aziz

The correlation is used for pressure loss, holdup, and flow regime. The Govier, Aziz &

Fogarasi correlation was developed following a study of pressure drop in wells producing gas

and condensate and is thus recommended for them. The new prediction method incorporates

an empirical estimate of the distribution of the liquid phase between that flowing as a film on


212

the wall and that entrained in the gas core. It employs separate momentum equations for the

gas-liquid mixture in the core and for the total contents of the pipe.

- BJA correlation

Baker Jardine & Associates have developed a correlation for two phase flow in gas-

condensate pipelines with a no-slip liquid volume fraction of lower than 0.1. This model

represents no major advance in theory, but rather a consolidation of various existing

mechanistic models, combined with a modest amount of theoretical development and field

data testing (PipeSim Manual). The model uses the Taitel Dukler flow regime map and a

modified set of the Taitel Dukler momentum balance to predict liquid holdup. The pressure

loss calculation procedure is similar in approach to that proposed by Oliemans, but accounts

for the increased interfacial shear resulting from the liquid surface roughness. The BJA

correlation is not recommended for systems having a non-slip liquid volume fraction greater

than 0.1. The quite extensive testing of the correlation against operating data has been

undertaken for horizontal and inclined flow, the test data for vertical flow is not so

comprehensive. (PipeSim Manual).

- Ansari:

The Ansari model was developed as part of the Tulsa University Fluid Flow Projects (TUFFP)

research program. A comprehensive model was formulated to predict flow patterns and the

flow characteristics of the predicted flow patterns for upward two-phase flow. The

comprehensive mechanistic model is composed of a model for flow pattern prediction and a

set of independent models for predicting holdup and pressure drop in bubble, slug, and

annular flows.

The basic plot and data fro calculations can be written to an Excel file by specifying one of

the Excel options for the output file format. This will then create an Excel data file called

with the extension .plt

Appendix E

HOLA 3.1 - Multiphase Flow Simulator

Appendix E – HOLA 3.1 - Multiphase Flow Simulator __________________________________________________________________________

214

E.1. Introduction

The geothermal steam-water flow flowing temperature and pressure profiles has been

simulated using HOLA 3.1 wellbore geothermal simulator. The simulator can also determine

the relative contribution of each feedzone for a given discharge conditions. The flow within

the well is assumed steady-state at all times, but time changing reservoir pressures are

allowed. Te HOLA 3.1 simulator uses several files in its computations. The some of them are

created by the program but others are read only and must be construed as a text file with DOS

acceptable names and with program acceptable format. HOLA also creates the calculations

and iterations output files. The first contains pressure and temperature profile including

related parameters change across the wellbore. The second provides information on the

iterations executed by HOLA in order to obtain the above output file. Plot of calculated and

measured pressure and temperature can be drawn by the program. This option enables to

view the results of the simulations.

The simulator can handle both single and two-phase flows in vertical pipes and calculates the

flowing temperature and pressure profiles in a well. It solves numerically the differential

equations that describe the steady-state energy, mass and momentum flow in a vertical pipe.

The code allows for multiple feedzones, variable grid spacing and radius.

The code was developed in the Fortran programming language by Lawrence Berkley

Laboratory, University of California. A detailed description of the formulation used in the

simulator is given in a separate report by Bjornsson (1987). A reference to this publication is

made frequently in the following text.


215

The simulator is consistent with SI units in all calculations except for pressure (bar instead of

Pa) and enthalpy (kJ/kg instead of J/kg).

Before downhole calculations are performed with the simulator HOLA 3.1, a definition of all

input parameters is required. The following text gives the program description, and also

shows the approaches that were used in order to solve the program deficiency problems and

calculate required parameters.

E.2 Governing Equations

The flow of fluid in a geothermal well can be represented mathematically by two sets of

equations. Between the feedzones, the flow is represented by one-dimensional steady state

momentum, energy and mass flux balances. When a feedzone is encountered, mass and

energy balances between the fluid in the well and the feedzone are performed. The solution of

these equations require fully defined flow conditions at one end of the system (inlet

conditions) and fully defined boundaries (wellbore geometry, lateral mass and heat flow).

The governing equations are then solved in small, finite steps along the pipe.

Whenever a feedzone is encountered, the mass and energy of inflow (or outflow) are known

and mass and energy balances performed, allowing for continuation of the calculations.

The governing steady-state differential equations for mass, momentum and energy flux in a

vertical well have already been given in chapter 5, I will bring them again here in simple

form:


216

0=dzdm (E.1)

accelevf dzdp

dzdp

dzdp

dzdp

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= (E.2)

QdzdEt ±= (E.3)

where m is the total mass flow, p is pressure, Et is the total energy flux in the well and z is the

depth coordinate. Q denotes the ambient heat loss over a unit distance. The plus and minus

signs indicate down flow and up flow respectively. The pressure gradient is composed of

three terms: wall friction, acceleration of fluid and change in gravitational load over dz. The

above terms were described in detail in chapter 5 which deals with multiphase flow in wells.

The governing equation of flow between the well and the reservoir is:

( )wfrG

GrG

L

LrLfeed pP

kkPIm −⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⋅+

⋅⋅=

µρ

µρ

(E.4)

where mfeed is the feedzone flowrate, PI is the productivity index of the feedzone, kr is the

relative permeability of the phases (subscripts L for liquid and G for steam), µ is the dynamic

viscosity, ρ is density, pr is the reservoir pressure and pwf is the flowing pressure in the well.

The relative permeabilities are here calculated by linear relationships (krG = S and krL = 1-S

where S is the volumemetric steam saturation of the reservoir). A flow into the well is

positive and flow from the well into the formation takes a negative sign (Bjornsson, 1987).


217

E.3 The computational models of HOLA 3.1

The simulator HOLA offers six modes of calculating downhole conditions in geothermal

wells. These are:

1. Outlet conditions known at the wellhead: The simulator calculates pressure,

temperature and saturation profiles from given wellhead conditions and given flowrates and

enthalpies at each feedzone except the bottom one.

2. Required wellhead pressure and multiple feedzones: The simulator finds the

downhole conditions that fulfill a required wellhead pressure. Also given are the productivity

indices, reservoir pressure and enthalpies at each feedzone. The feedzones must have a

positive flowrate.

3. Required wellhead pressure and two feedzones: This mode is similar to mode 2,

except that only two feedzones are allowed and each can either accept or discharge fluid.

4. Required wellhead flowrate and two feedzones: In this mode, the simulator finds

downhole conditions that fulfill a required wellhead flowrate. Only positive flow is allowed

from the feedzones.


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5. Required wellhead injection rate and two feedzones: The simulator iterates for the

downhole conditions that provide the required wellhead injection rate. Only two feedzones

are allowed and both must accept fluid. Thus has negative flowrates in the program.

6. Variations in wellhead pressure and enthalpy for a constant flowrate and given

reservoir pressure history at two feedzones: This mode is similar to mode 4, except that now

a history is specified for the reservoir pressure. Only two feedzones are allowed and both

must discharge to the well.

E.4 Heat loss parameters

The simulator handles heat transfer between the well and the reservoir by formulation given

by Bjornsson (1987). The heat loss parameters necessary are:

- rock thermal conductivity

- rock densities

- rock heat capacity

- time passed from initial discharge

These parameters specify the thermal conductance to the ambient rocks. The parameters used

in present work have been taken from Prats SPE paper (2001). His work treats about steam

injected to the reservoir, but the heat transferred from steam to the reservoir goes on the same

rules like for geothermal wells, and the typical rock properties presented in his work given in


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the appendix (7.1.) are assumed to be appropriate for these simulations. 2 W/m/°C, 2800

kg/m3, are used for the rock thermal conductivity and density respectively. The heat loses in

this option may also be neglected when the thermal conductivity value is set to be zero in this

option.

In addition a lateral temperature gradient between the well and the ambient may be

established defining a reservoir temperature curve from the surface to the feedzone level. The

simulator requires in this option the temperature value at given depth and then interpolates

linearly between these data points in order to evaluate the formation temperature gradient.

E.5 Wellbore geometry

The wellbore geometry need to be specified. The program allows for two different tubing

diameter segments. The inflow is interpreted to be at the bottom of the deeper section, thus in

the cases where the last feedzone does not occur at the well bottom the apparent depth need

to be set equal to the depth of the lowest feedzone. The well inner radius and wall roughness

need to be specified. The well roughness in present work was assumed to be 0.024 mm, and

was taken from the PipeSim Table E.1.

The downhole computations proceed along the well in finite-difference steps of ∆z. After

each step the flowing conditions of the well are calculated at the subsequent nodal points

defined in the program. The nodal points divide well into depth segments where parameters

of the well like diameter or feedzones may change. The distance between the nodal points are


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defined by the user. More than 25 m distance is suggested by the program authors due to

computer speed, however current PC handle the calculations for 1m nodal distance with few

seconds.

E.6 Feedzone properties

The feedzone properties that must be specified in the program depends on the computational

models selected. In mode 1 only the number of the feedzones, flowrate and the enthalpy of

the each feedzone except the bottom one must be specified. The properties of the bottom

feedzone, by conservation principles, are simply the residuals of the wellhead energy and

mass flow minus energy and mass flow of all the other feedzones. Using the other program

modes additionally productivity index, enthalpy and pressure at the feedzone need to be

entered A special care must be taken in specifying the feedzone enthalpies or one may end up

with negative temperatures in the well.

In present work only one feedzone was assumed for calculation, and some problem has

occurred due to the mode 4 requires two or more feedzones to be specified. The problem was

solved by placing both feedzones with a short distance of 20 m and assuming the same

conditions for both feedzones. Such system may be assumed to work as the one feedzone.


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E.7. Velocities of individual phases

The simulator HOLA 3.1 offers four different methods of calculating the average cross

sectional velocities of steam and water in a flowing well. Four different methods of

calculating the average cross sectional are available in the program. All of the methods have

their limitations and it is difficult to know which of them is best for a certain well.

Experience in using wellbore simulators and access to downhole data in the well under

consideration will often provide an appropriate selection of the phase velocity method.

In present work the Orkiszewski correlation was used. The description and the

recommendation for this choice are given in chapter 5, which deals with the multiphase flow.

The other correlations available are Armand, Chisholm and Bjornsson (for the 9 5/8” wells)

The Armand relations are semi empirical and based on limited experimental data, collected in

small diameter pipes at low flowrates (Bjornsson, 1987). The Chisholm in his correlation

proposed the relation for slip in the form

G

LL

G

xx

xx

kρρρ

ρ

⋅−

⋅+

⋅−

+⋅+=

14.01

14.016.04.0 (E.5)

This formula was modified in program for low pressure and low mass saturation of steam.

Existing downhole pressure profiles were used to estimate the slip ratio and a correction


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factor to the Chisholm formula defined. A care should be taken in the use of this relation

since it still needs validation (Bjornsson, 1987).

E.8 Productivity Index estimation

The IPR curves were calculated for the wells examples under considerations in this work in

order to asses the performance and fin the well operating point for two tubing diameter sizes.

This requires productivity index to be known for particular well that is the mathematical

means of expressing the ability of a reservoir to deliver the fluids to the wellbore. The

performance relationships are presented in Chapter 3. and here only the technical approach

used in order to determine the productivity index is described.

HOLA 3.1 simulator has not a detailed mode to calculate the productivity index, and the

productivity index was estimated from number of repeated simulations made in mode 2. The

wellhead parameters including temperature and flowing pressure and reservoir average

pressure was known. The enthalpy in this mode may be defined only at the feedzone thus in

the first step the enthalpy was adjusted from several simulation to arrive at the appropriate

wellhead temperature. Then initial guess for the productivity index was entered and the

obtained wellhead mass flowrate was compared with required. Increasing or decreasing the

productivity index results in estimation of the productivity index. Such approach was

necessary in present work, because there was no specific data about the reservoir parameters

available.

The vertical lift performance (VLP) curves for geothermal wells, was also calculated from

HOLA 3.1. The downhole flowing pressure pwf was calculated several times for the same


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wellhead pressure but different flowrates with the 2 kg/s step. The curves revealed to have the

unstable shape probably due to a problem with the simulator, e.g. because of flow regimes

and/or fluid properties. The curve fitting application in MATHLAB 6.5 program allowed to

get rid of the curves fluctuations and draw the plots.


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Table E.1 Thermal Conductivities

Material Density Thermal Conductivity Thermal Conductivity

(kg/m3) Btu/hr/ft/F (W/m/K)

Anhydrite 0.75

Carbon Steel 7900 28.9 50

Concrete Weight Coat 2000 - 3000 0.81 - 1.15 1.4 - 2.0

Corrosion Coat (Bitumen) - 0.19 0.33

Corrosion Coat (Epoxy) - 0.17 0.30

Corrosion Coat (Polyurathane) - 0.12 0.20

Dolomite 1.0

Gypsum 0.75

Halite 2.8

Ice 900 1.27 2.2

Lignite 2.0

Limstone 0.54

Line pipe 27 46.7

Mild Steel tubing 26 45

Mud 1500 0.75 - 1.5 1.3 - 2.6

Neoprene Rubber - 0.17 0.3

Plastic coated pipe 20 34.6

Plastic coated tubing 20 34.6

Polyurathane Foam (dry) 30 - 100 0.011 - 0.023 0.02 - 0.04

Polyurathane Foam (wet) - 0.023 - 0.034 0.4 - 0.6

PVC Foam (dry) 100 - 340 0.023 - 0.025 0.040 - 0.044


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Sandstone 1.06

Shale 0.7

Stainless Steel - 8.67 15

Stainless steel (13%) 18 31.14

Stainless steel (15%) 15 26

Syntactic Foam (dry) 500 0.052 0.09

Syntactic foam (wet) - .017 0.3

Volcanics 1.6

Wet Sand 1600 1.04 - 1.44 1.8 - 2.5

Table 7.2 Roughness Material ft. in

Drawn tubing(brass, lead, glass, and the like) 0.000005 0.00006

Commercial steel or wrought iron 0.00015 0.0018

Asphalted cast iron 0.0004 0.0048

Galvanized iron 0.0005 0.006

Cast iron 0.00085 0.010

Wood stave 0.0006-0.003 0.0072-0.036

Concrete 0.001-0.01 0.012-0.12

Riveted steel 0.003-0.03 0.036-0.36

Appendix F

Simulations results in oil wells

metering of two-phase geothermal wells using pressure pulse technology

Documents

metering of multiphase

multiphase flow models

flow patterns

multiphase flow

oil wells

pressure pulse method

pressure surge

pressure gradient