introduction to nodal analysis (1)
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Introduction to NODALanalysis
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Learning Objectives
Inflow Performance Relationship (IPR) Single phase
Two phase
Vertical Lift Performance Single phase
Two phase
Flow Through Chokes
Matching Inflow and Tubing Performances
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Introduction
Production by natural flow Need for better understanding of various concepts
which define well performance. Pressure loss occurs in:
the reservoir the bottom hole completion the tubing or casing the wellhead the flowline the flowline choke pressure losses in the separator and export pipeline to
storage
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Introduction Production is generally limited by the pressure in the reservoir
and difficult to do something about it. A major task is to optimise the design to maximise oil and gas
recovery.
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Production Performance
Production performance involves matchingup the following three aspects: Inflow performance of formation fluid flow from
formation to the wellbore.
Vertical lift performance as the fluids flow up thetubing to surface.
Choke or bean performance as the fluids flowthrough the restriction at surface.
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Fluid Flow Through Porous Media
The ability to determine the productivity of a reservoirand the optimum strategy to maximise the recoveryrelies on an understanding of the flow characteristicsof the reservoir and the fluid it contains.
The interaction between the fluid (and its properties)
and the rock (and its properties) Comparison with flow through pipes.
Multiple fluids Surface tension
Capillary forces
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Fluid Flow Through Porous Media
The nature of the fluid flow Time taken for the pressure change in the reservoir Fluid to migrate from one location to another For any pressure changes in the reservoir, it might
take days, even years to manifest themselves inother parts of the reservoir.
Therefore flow regime would not be steady state Darcys law could not be applied Time dependent variables should be examined
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Two Phase Flow, Vogels Equation
2
r
wf
r
wf
maxo
o )P
P(8.0)
P
P(2.01
q
q=
A simplified solution was offered by Vogel. He simulated the PVT
properties and cumulative production from different wells on computer to
produce many IPR curves. These were then normalised for pressure and
producing rate. The curves produced represent many different depletion
drive reservoir. A single curve can be fitted to the data with the following
equation.
This equation has been found to be a good representation of many
reservoirs and is widely used in the prediction of IPR curves for 2-phaseflow. Also, it appears to work for water cuts of up to 50%.
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Vogels Equation, Example-1
b/d211)2400800(8.0)
2400800(2.01250)
PP(8.0)
PP(2.01qq
psi800PFor
b/d250
)2400
1800(8.0)
2400
1800(2.01
100
)P
P(8.0)
P
P(2.01
qq
psi1800P
b/d100q
psi2400P
:datafollwoingthegivenpsi,800PforqandqFind
22
r
wf
r
wfmaxoo
22
r
wf
r
wf
omaxo
wf
o
r
wfoomax
= = =
=
=
=
=
=
=
=
=
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Vogels Equation, Example, Cont.
If other values of Pwf
are chosen, sufficient
qos can be generated
to plot the curve, e.g.:
Pwf qo800 211
1200 175
1600 128
2000 69
IPR
0
500
1000
1500
2000
2500
3000
0 100 200 300qo
Pwf
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Vogels Equation, Combined Single Phase Liquid
and 2-Phase
In this case there is a single
phase liquid which exists
above the bubble point. Below
the bubble point the system
becomes 2-phase.
The figure opposite shows the
IPR, which is a combined
linear-Vogel plot (i.e., straight
line above Pb and Vogel
below Pb with Pb substitutedfor Pr).
Pb
Pr
qb qmaxq
Pwf
Straight line above Pb
Vogel below Pb
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Vogels Equation, Example-2
psia1000b.psia2500a.:ofPforqiii)
PbelowIPRVogelassuming,q)qi)
:Find
)4
3(ln
)(1008.7
cp0.682.1B0S
ft0.4rft2000rft60h
md30kpsia2000Ppsia3000P
:datafollwoingGiven the
wfo
bmax
b
3
o
we
b
ii
r
rB
PPhkq
w
eoo
wfsro
o
o
r
=
===
======
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Example-2, Solution
=
=+
=
=
2
max
b
3
3
)(8.0)(2.01
PbeyondVogelusingii)
b/d2010
)04
3
4.0
2000(ln2.168.0
)20003000(60301008.7
)
4
3(ln
)(1008.7
:usedisequationinflowradialforethere
point,bubbletheabovePIgivennoisTherei)
r
wf
r
wf
oo
w
eoo
wfsro
o
P
P
P
Pqq
r
rB
PPhkq
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Example-2, Solution
b/d/psi01.220003000
2010PatPItherefore
8.1
PPI)Vogel(q
P
8.1q
P
P6.1
P
2.0q
dP
qd-PIPPatand
P
P6.1
P
2.0q
dP
qd-
P
P6.1
P
2.0q
dP
qd
PI.thegivesitateddifferentiisequationsVogel'ifIPR,theofslopetheisPIthethatmemberingRe
b
bmaxo
b
maxo2b
b
b
maxo
wf
obwf
2
r
wf
rmaxo
wf
o2
r
wf
rmaxo
wf
o
=
=
=
=
+===
+=
=
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Example-2, Solution
b/d357315632010qqq
b/d1563)200010000.8()
20001000(2.01qq
Pi.e.psi,1000Pb.
b/d1005)25003000(01.2)PPPI(q
,Pi.e.psi,2500Pa.iii)
b/d424322332010qqq
b/d22338.1
200001.28.1
PPIq
o(Vogel)bo(total)
2)Vogelmax(o(Vogel)
bwf
wfr
bwf
)vogelmax(b)totalmax(
b)vogelmax(o
=+=+=
= =
=
=+=+=
===
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Vogels Equation, Problems-1&2
IPRthePlot
b/d/psi2PIpsi3000P
psi4200P
psi.2500Pforqand,q,qfinddata,followingtheUsing
2-Problem
_______________________________________
psig1000Pb/d150q
psig1600Ppsig1600P
:datafollowingtheforIPRplotandqFind
1-Problem
b
r
wfmax(total)b
wfo
br
omax
==
=
=
==
==
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Two Phase Flow: Effect of GOR
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Non Darcy Flow
Darcys law only applies to laminar flow situations, a valid
assumption for the majority of oil wells. For gas wells and some very high flowrate (light crude) oil
wells, the volumetric expansion as fluid approaches the
wellbore is very high and this can result in turbulent flow.
In such cases, a modified form of the Darcy equation,
known as the Forchheimer equation, is used:
2U
K
U
dr
dP+
=
The non-Darcy component due to turbulent flow is
normally handled as an additional pressure loss PND
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Productivity Index (PI)
Productivity index is a measure of the capability of a
reservoir to deliver fluids to the bottom of a wellbore.
It relates the surface production rate and the pressure drop
across the reservoir, known as the drawdown.
To take into account the effect of the thickness of producing
interval and comparison of various wells, the Specific
Productivity Index is defined as:
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PI For SS Incompressible Flow
PI is constant if, B and K remain constant. Plot of Pwversus qs should be a straight line of slope 1/J, with
an intercept on the ordinate axis of Pe.
PI for Semi-Steady State Incompressible Flow
)rrln(B
Kh10082.7
PP
q
J)PI(
w
e
3
we
s
SSSS
===
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Oil Wells Productivity Index
The Productivity Index (PI)is the ratio ofproduction to the pressure draw down at the mid-point of the production interval
rateflowoilQpresureflowingP
presurestaticPPP
QPI
owf
wiwfwi
o
==
=
=
The productivity index is a measure of the oil well potential or ability
to produce and is a commonly measured well property.
PI is expressed either in stock tank barrel per day per psi or in
stock tank cubic metres per day per kPa.
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Practical determination of PI
The static pressure (Pwi
) is measured by:
prior to open a new well (after clean up)
after sufficient shut in period (existing wells)
In both cases a subsurface pressure gauge is run into
the well
The flowing bottom hole pressure (Pwf) is recorded
after the well has flowed at a stabilised rate for a
sufficient period (new wells)
prior to shut in for the existing wells
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Decline of PI at High Flow Rates
In most wells the productivity index remains
constant over a wide range of variation inflow rate. Therefore, the oil flow rate isdirectly proportional to bottom hole
pressure draw down.
However, at high flow rate the linearity failsand the productivity index declines, whichcould be due to:
1- turbulence at high volumetric flow rates
2- decrease in relative permeability due tothe presence of free gas caused by the dropin pressure at the well bore
3- the increased in oil viscosity withpressure drop below bubble point
Flow rate
PI
Drawdown
Qo PI
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Example 1
A well has a shut in bottom hole pressure of 2300 psia and
produces oil at 215 barrels/day under a draw down of 500 psi.
The well produces from a formation of 36 feet net productive
thickness. What is productivity index, and specific productivity
index?
Specific productivity index
Productivity Index is a function of productive thickness (in fact,the length of perforation interval). In order to compare thewells with each other, the specific productivity index (PI)s isdefined as:
ions)(performatzonepaytheoflengthh
)PP(h
QhPI)PI(
wfwi
os
=
==
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Tubing Performance
The pressure loss in the tubing can be a significant
proportion of the total pressure loss. However its
calculation is complicated by the number of
phases which may exist in the tubing.
It is possible to derive a mathematical expressionwhich describes fluid flow in a pipe by applying
the principle of conservation of energy.
The principle of the conservation of energyequates the energy of fluid entering in and exiting
from a control volume.
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Fundamental Derivation of Pipe Flow Equation
The principle of the conservation of energy equates
the energy of fluid entering and exiting from a controlvolume.
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Flow Regimes in Vertical 2-Phase Flow
As the pressure on a crude oil containing gas in solution is
steadily reduced, free gas is evolved and as a consequence, the
liquid volume decreases.
This phenomenon affects the relative volumes of free gas andoil present at each point in the tubing of a flowing well.
If the bottom hole pressure in a well is above the bubble point
of the crude oil, single phase liquid is present in the lower part
of the tubing.
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Flow Regimes in Vertical 2-Phase Flow, Cont.
As the liquid moves up the tubing, thepressure drops and gas bubbles begin toform. This flow regime where gas bubblesare dispersed in a continuous liquidmedium is known as bubble flow.
As the fluid moves further up the tubing,
the gas bubbles grow and become morenumerous. The larger bubbles slip upwardat a higher velocity than the smaller ones,because of the buoyancy effect.
Single Phase
Liquid Flow
Bubble
Flow
Slug or Plug
Flow
Annular
Flow
Mist
Flow
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Flow Regimes in Vertical 2-Phase Flow, Cont.
A stage is reached where these large bubbles extendacross almost the entire diameter of the tubing. As aresult, slugs of oil containing small bubbles areseparated from each other by gas pockets that occupythe entire tubing cross section except for a film of oilmoving relatively slowly along the tubing wall. This isSlug or Plug Flow.
Still higher in the tubing, the gas pockets may havegrown and expanded to such as extent that they areable to break through the more viscous oil slug. Gasforms a continuous phase near the centre of the tubing
carrying droplets of the oil up with it. Along the walls ofthe tubing there is an upward moving oil film. This isAnnular Flow.
Single Phase
Liquid Flow
Bubble
Flow
Slug or Plug
Flow
Annular
Flow
Mist
Flow
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Flow Regimes in Vertical 2-Phase Flow, Cont.
Continued decrease in pressure with resultantincrease in gas volume results in a thinner and thinneroil film, until finally the film disappears and the flowregime becomes a continuous gas phase in which oildroplets are carried along with the gas, i.e., Mist Flow.
Not all these flow regimes will occur simultaneously ina single tubing string, but frequently 2 or possibly 3may be present.
In addition to flow regimes, the viscosity of oil and gasand their variation with pressure and temperature,
PVT characteristics, flowing bottom hole pressure(BHP), and tubing head pressure (THP) affect thepressure gradient. Single Phase
Liquid Flow
Slug or Plug
FlowBubble
Flow
Annular
Flow
Mist
Flow
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Flow Regimes in Vertical 2-Phase Flow, Cont.
These flow patterns have been observed by a number ofinvestigators who have conducted experiments with air-water
mixtures in visual flow columns.
The conventional manner of depicting the experimental data
from these observations is to correlate the liquid and gas
velocity parameters against the physical description of theflow pattern observed.
Such presentations of data are referred to as flow pattern
maps. The map is a log-log plot of the superficial velocities
of the gas and liquid phases.
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Practical Application of Multiphase Flow
Multiphase flow correlations could be used for: Predict tubing head pressure (THP) at various rates
Predict flowing bottom hole pressure (BHP) at various rates
Determine the PI of wells
Select correct tubing sizes Predict maximum flow rates
Predict when a well will die and hence time for artificial lift
Design artificial lift applications
The important variables are: tubing diameter, flowrate, gas liquid
ratio (GLR), viscosity, etc.
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Liquid-Liquid Flow
The case of liquid-liquid flow in production wells may
occur in low GOR wells which produce water.
Since both phases are only slightly compressible or
incompressible, it would be expected that the physical
nature of the flow of an oil-water mixture to surface
would not be as dramatically different from single phaseliquid flow as the oil-gas system.
If oil and water enter the wellbore from the reservoir and
flow up the tubing to surface, the physical distribution of
the phases will depend upon their relative volumetricproperties, ie, one phase will be continuous and the other
dispersed.
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Liquid-Liquid Flow
Unlike the gas therewill be little relativevolumetric expansion
between the twophases.
Thus, the physicaldistribution will bemore dependent onthe WOR and the
flow velocity.
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Pressure Transverse or Gradient Curves
A, B, C=DifferentTubing HeadPressures
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Pressure Transverse or Gradient Curves
By shifting the curves
downwards, he found that, fora constant GLR, flowrate andtubing size, the curvesoverlapped
Then, a single curve could beutilised to represent flow in thetubing under assumedconditions.
The impact was in effect toextend the depth of the well bya length which, would
dissipate the tubing headpressure. A, B, C=Different
Tubing HeadPressures
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Gradient Curves
Gilbert was then able to
collect all the curves for aconstant tubing size and
flowrate on one graph,
resulting in a series of
gradient curves whichwould accommodate a
variety of GLRs.
He then prepared a series
of gradient curves atconstant liquid production
rate and tubing size.
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Gradient Curves
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Positive or Fixed Choke This normally consists of two
parts: A choke which consists of
a machined housing into
which the orifice capability
or "bean" is installed.
A "bean" which consists of
a short length 1-6", of thick
walled tube with a smooth,
machined bore of specified
size.
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Valve Seat with Adjustable Valve Stem
In this design, the orifice
consists of a valve seatinto which a valve stemcan be inserted andretracted, thus adjusting
the orifice size. The movement of the valve
stem can either be manualor automatic using anhydraulic orelectrohydraulic controller.
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Choke Flow Characteristics
Chokes normally operate in multiphase
systems. Single phase can occur in dry gaswells.
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Critical Flow through Chokes
R=P2/P1The value of R at the
point where the
plateau production
rate is achieved istermed the
critical pressure ratio
Rc.
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Critical Flow through Chokes
Critical flow behaviour is only exhibited by highly
compressible fluid such as gases and gas/liquid mixtures. For gas, which is a highly compressible fluid, the critical
downstream pressure Pc is achieved when velocity
through the vena contracta equals the sonic velocity this means that a disturbance in pressure or flow
downstream of the choke must travel at greater than thespeed of sound to influence upstream flow conditions.
In general, critical flow conditions will exist whenRc=
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Multiphase Flow through a Choke A number of researchers have published studies on
multiphase flow through chokes. Some of the studies relate to correlation of field
measurements.
PTH = tubing head flowing pressure in psia
Cd = constant
R =gas liquid ratio (MSCF/bbl)Q =oil flowrate (STB/d)
S =bean size in 1/64"
Gilbert (435 is correct)
Achong (R in SCF/bbl)
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Matching the Inflow and Tubing Performance
Method 1 - Reservoirand tubing pressure loss
convergence in
predicting bottomhole
flowing pressure
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Matching the Inflow and Tubing Performance
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