1 Productivity
Figure 1 Items in the production system
Figure 1 illustrates way from the reservoir, through well and flow line, till the process platform.
Production means that the oil / gas flows from the reservoir rock into the well, from bottom to
the top of the well and further flow line riser and pipe; through regulation valve into the process
plant. Since the flow through the reservoir, pipe and the valve is governed by different laws of
physics, it is natural to distinguish between
-- Flow from the reservoir to the well
-- Flow in well pipe and flow line
-- Flow through regulation valve
The production rate is determined by the combination of these elements and the inlet and outlet
condition, in other words, the
-- Reservoir and the reservoir fluid
-- Process inlet pressure, and other restrictions in the process
To predict production, it is necessary to determine the relationships between pressure and rate for
the elements above and combine them. Since the relationships depend on well design, we will
then also be able to calculate the well design that appear to give the best productivity; in other
words, optimize well design and completion.
In Chapter 1, we will use simple relations for the influx, flow in the well and through regulation
valve. The purpose is to show how these relations can be combined to predict production.
1.1 To the flow
In order for a well to produce, fluid must be drawn from the reservoir. Pressure drop in reservoir
and well represent loss of energy, because of the flow resistance. The relation between bottom
hole pressure and rate is the starting point to construct a production system based on the natural
characteristics of the reservoir. We can measure this relation directly, or predict it out from
reservoir properties.
1.1.1 Linear flow characteristics
Figure 1.1 indicates measured pressure and rates and personalized relationship between them.
The linear context indicated, we can express that
pw  p R 
1
qo
J
(1-1)
J: productivity index (Sm3/d/bar) J  qo  pR  pw 
pR: reservoir "pressure" (bar)
pw: well potential "pressure" (bar)
qo: production (Sm3 / d)
In chapter 2, we estimate productivity index from Darcy's law and the reservoir. Mathematical
relationships makes it possible to predict how productivity index depends on the well design. But
because we rarely know the conditions down in the reservoir, well, you should mathematical
estimates verified with the measurements.
Challenges
Estimate out from Figure 1.1:
-- Reservoir and productivity indeks (answer: pR = 220 bar, J = 20 Sm3 / d).
-- If we lower the production well to the 175 bar?
-- Production of this well, when the reservoir has decreased to 200 bar?
Figure 1.1: Inflow characteristics
What determines the inflow characteristics?
Inflow characteristics quantify flow resistance from the reservoir to the well. Flow resistance in
porous materials we usually quantified by Darcy's law: v   k   dp ds where v is the volume
flow divided by the cross section: v  Q / A  qB / A . From this we can express pressure loss
from the outer boundary of the well
re
pe  pw  
rw
 qB
k A
ds  q
B 1
re
k
 A ds
(1-2)
rw
Where "s" indicates the flow line; by radial flow it would be a straight line from the outer radius
of the well. Viscosity and volume factor depend on the fluid. For oil, the product: oBo is usually
not much affected by pressure such that it is considered constant along the flow lines. By
combining (1-1) and (1-2), we can express the productivity index
Je 
q

pe  pw
k
re
 B  A1ds
rw
The area: A depends on the geometry, but not pressure, at least not for incompressible fluids.
While Darcy's law applies, the flow geometry will not depend on rate. Thus, although the area A,
will vary with location, but is unaffected by pressure and rate. The height of the reservoir layer
can often be considered constant: A = hS (s). Productivity index can then be expressed in general
Je 
k
re

 B  A ds
1
kh
f  geometri 
B
(1-3)
rw
The production index will thus depend on strømningsgeometrien (reservoir geometry, well
location, complemented length) and fluid viscosity, but these parameters are hardly sensitive to
pressure changes. The permeability depends on the pore structure of the rock.
When the pressure drops, the flow channels collapse. Chemical precipitates can clog. This will
then reduce the productivity index. However, while the rock structure will not change the
productivity index be constant.
1.2 Wellbore flow
1.2.1 Flow in the production pipe
Production pipe should be selected in accordance with the natural characteristics of the well. The
basis for such selection is pressure and flow computation by the pipe flow equation
dp  g x dx  vdv 
1  2
f v dx  0
2 d
(1-4)
fluid density (kg/m3)
v: flow velocity (m / s)
x: length along the pipe, the flow direction (m)
gx : component of gravity acceleration in the flow direction (m/s2)
f : the friction factor (-)
For single-phase oil flow in the pipe with a constant diameter, it can be assumed that the speed is
constant, thereby neglecting vdv. The speed depends on the flow rate and pipe diameter
v
qo Bo
 d2 4
Too large velocity could accelerate corrosion, in extreme cases also cause mechanical erosion.
Too small velocity could lead to accumulation of produced sand. By integrating (1-4) along the
production pipe (length: L), we can estimate the pressure valve on the tree; peak pressure
pth  p w  g x L 
1  2
f v L
2 d
(1-5)
1.2.2 Natural lift
Tubing head pressure characteristics specify how the pressure at the valve tree vary with rate:
pth(qo). For linear the flow characteristics and single-phase oil, we can predict tubing head
pressure from (1-1) and (1-5). We usually present tubing head pressure characteristics
graphically, as illustrated in Fig 1.3.
Challenges
Estimate out from Figure 1.3
-- Tubing head pressure at zero production
(answer: about 70 bar)
-- Tubing head pressure when the well is producing 200 Sm3/d
(answer: about 60 bar)
-- Production with separator inlet pressure: 20 bars?
(answer: about 620 Sm3/d)
-- The maximum rate with ruptured flow line
(answer: about 800 Sm3/d)
Figure 1.3 Tubing head pressure characteristics
A goal is to construct the well to achieve high production capacity. It means to avoid
unnecessary flow restriction and pressure loss. Relations above make it possible to predict the
effect of: productivity index, pipe diameter, the friction factor, separator pressure, will have on
production capacity.
Challenge
Estimate, according to Figure 1.3
-- Production by reducing the separator pressure from 20 to 10 bar
(answer: about 730 Sm3 / d)
1.2.3 Downhole pumping
To increase production downhole pump can be installed. This means that the inlet pressure to the
production pipe now becomes equal to the well pressure plus pressure increase provided by the
pump. When the pump can be regulated, choking before the separator is not necessary.
It is practical to predict the change in pressure from the separator and down into the inlet of the
production pipe. The pipe equation for single-phase flow provides
pti  ps  g x L 
1  2
f v L
2 d
(1-6)
Pressure downhole will as before depend on the pressure loss by flow from the reservoir to the
well (1-1), ie: pw  pR  1 J qo . The pump must increase the pressure: p p  pti  pw , to
maintain the flow. This can be expressed graphically by downhole characteristics below.
Figure 1.4: Downhole characteristics
Downhole well pressure: pw is provided by reservoir (reservoir pressure minus inflow pressure
loss). Inlet pressure to the production pipe: pti is required to lift the production to the inlet
separator. The pump must provide the difference: pp = pti - pw.
Energy demand is usually expressed by first law of thermodynamics: dE = q-w. The pump
provide mechanical work: w. Heat transfer between the pump and the environment will be
negligible: q ~ 0. The flow equation (1-4) is a mechanical force balance (pressure=force/area).
By expressing the fluid density as the ratio between the mass flow and volume rate:   m / Q ,
we can transfer (1-4) to energy units
dE  Qdp  mg x dx  mvdv  m
1 f 2
v dx
2d
Over a pump will the speed and height changes are negligible, if any. For a frictionless pump:
dEi  Qdp   w , the thermodynamically effect becomes
pti
Ei 
 Qdp  Qp
pw
p
(1-7)
Pumps often have considerable friction losses. Usually we consider the thermodynamic effect as
basis and correct for friction by with an "efficiency factor", for the current pump.
E  Q p p 
1.3 Rate regulation
To regulate the rate of individual wells, we use the choke valves mounted after the valve tree.
Pressure reduction is acieved leading a fluid jet into a wider pipe cross-section. By this, the
velocity energy is dissipatdby turbulence between the jet and the fluid around it. Pressure loss
due to the wall friction will usually be negligible. Figure 1.4 illustrates the simplest construction
of such a valve
Figure 1.4 Flow through simple nozzle
The behavior can be quantified by combination of the Bernoulli's equation and Carnot
relationship. Bernoulli's law describes the pressure until the end nozzles. For incompressible
fluid, this gives
pc  pth 

1
 vc2  vth2
2

pc: outlet pressure
pth: inlet pressure, tubing head pressure
vth: flow speed before the nozzles
vc: speed at the end of the nozzles
Bernoulli's law ignores friction losses. When the nozzles stream flows out into nearly stagnant
fluid, turbulence leads to large friction losses; so that Bernoulli's no longer applies. After the
nozzle the following balance (Carnot relationship) of pressure forces and linear momentum
applies
m vs  m vc    ps  pc A  0
 : Flow rate
m
A: pipe cross-section
vs: flow speed for the nozzles, inlet speed of the separator
By: m   vs A , into the relationship above, the downstream pressure is expressed as

ps  pc   vs vc  vs2

The combination of this and (1-6) has given the pressure reduction over the choke valve:
pth  ps 

 
1
 vc2  vth2   vs vc  vs2
2

The first term of the relation aabove provides the pressure reduction due to acceleration through
the nozzles (Bernoulli). The second term provides pressure recovery after the outlet (Carnot ).
We assume little compressible fluid and equal pipe size before and after the nozzles, so that the
velocites will be the same: vth = vs. The pressure reduction can be expressed
2
A 
1 Q 
1 Q
pth  ps      1  c     
2  Ac  
A
2  Ac 
2
2
(1-8)
The simplifying indicated above (1-7) assumes the opening through the valve is much smaller
than the pipe cross-section: Ac <<A. In valves designed for pressure reduction, this will usually
be the case
Figure 1.6 The production rate controlled by the regulation valve
Today choke valves are often designed such that jets rays collide with each other. Openings are
often adjustable.
In Venturi nozzles, the cross section changes gradually. This limits the turbulence, so that most
of the velocity energy is reclaimed as pressure and Bernoulli's law applies. Then the pressure
after the nozzles is almost the same as the upstream pressure. Such nozzles will not substantially
affect the rate.
1.5 Future production
With time, the reservoir conditions will change, depending on production and injection. Such
changes affect
-- Future production and income
-- How different production decisions will affect future production
Future pressure and fluid composition are usually forecasted by numerical reservoir simulation.
Below a reservoir model is simplified sufficiently to be analytical solvable. Combining this with
well flow relation provides fundamental insight into the dynamics of field production.
1.5.1 Reservoir model
We will assume that the reservoir is producing by expansion, quantify by the compressibility
equation
c
1 dV
V dp
c: fluid compressibility, under reservoir conditions
V: fluid volume in the reservoir
dp: pressure reduction
dV: additional fluid volume, because of the pressure reduction
Reservoir fluids fill the pore space. Thus,”additional fluid volume” corresponds to production.
From the compressibility equation, we can relate production to change in reservoir pressure

dpR
1 dV
1


qo Bo
dt
cV dt cV
(1-9)
Numerical reservoir simulation may solve the relationship above in 3-dimentions. Here, we
ignore pressure gradients in the reservoir and assume constant pore volume and constant fluid
compressibility. The relationship above may then be solved analytically to provide pressure
change due to production
pR ( to )  pR ( t ) 
1 t
 qo Bo dt
cV to
(1-10)
pR: average reservoir pressure
qo: production rate
Equation (1-8) can be viewed as a simple reservoir model. By solving for different assumptions
about the flow and production, we will be able to predict future production. This is shown below
1.5.2 Plateau
Processing capacity: qop, may initially limit the production. With production rate equal
processing capacity, we can solve (1-8). This shows that under these assumptions, the reservoir
pressure declines steadily with time
pR t   pRi 
Bo
qop t
cV
(1-11)
With declining reservoir pressure, the well pressure will also decline, as illustrated by figure 1.6.
Constant production may be maintained by gradually opening the choke valve; until the tubing
head pressure equals the inlet separator pressure: pth = ps. After that, we will no longer be able to
maintain a constant rate.
Challenges
From figure 1.6 predict
-- Pressure decrease across the choke valve , year zero
-- Length of the plateau
Figure 1.6 Pressure Drop at constant rate,
(assumed constant compressibility and reservoir volume)
1.5.3 Declining production
We consider fully open choke valve, providing constant tubing head pressure. With pressure loss
along the pipe dominated by the static component, the bottom well pressure will then also be
approximately constant. It then follows from (1-1) that the rate change must then correspond to
the change in reservoir pressure
dqo   J d pR
Setting this into the reservoir relation (1-9) and integrate, predict the production profile after the
plateau period
qo t   qop e


J Bo
t t p
cV

(1-12)
Under the assumptions above, production would decline exponentially, as illustrated in Figure
1.7 below. Equation (1-12) also makes it possible to predict how changes in the reservoir and
production conditions will affect the immediate and future production.
Figure 1.7 Declining production
Figure 1.8 illustrates the pressure over time, based on relationships and conditions above
Figure 1.8 Future pressures, according to the simple reservoir model
1.7 Literature
Gilbert, WE: Flowing and Gas-Lift Well Performance
Paper A.P.I. 801-30H. Presented at the spring meeting of the Pacific Coast District, Division of
Production, API, Los Angeles May 6-7, 1954
Golan, M., and Whitson, C.H. : Well Performance
IHRDC, 1986, Prentice-Hall, 1991
Economides, MJ, Watters, LT, Dunn-Norman, S.: Petroleum Well construction
Wiley, 1998