Transient pressure in long wellbores
Pål Skalle
Department of Petroleum Engineering and Applied Geophysics, NTNU, Trondheim, Norway
pal.skalle@ntnu.no
Tommy Toverud
Department of Petroleum Engineering and Applied Geophysics, NTNU, Trondheim, Norway
ttoveru@hotmail.com
Abstract
Drilling of long oil wells introduces a “new” challenge; after pump start the generated pump pressure, i.e. the hydraulic
fiction resistance in the circulation system, needs a transient period of several minutes before it reaches the steady state pressure.
A theoretical model to estimate the transient pressure was developed, on basis of classical water hammer theory (pressureincrease due to sudden valve closure in a flowing pipe line). The model was compared with observed transient periods in long
wellbores, and the model and fitted well with observations.
This knowledge is important when analyzing pressure testing and other sudden pressure change-situations. The transient
pressure behavior masks the true pressure during the transient period.
1. Introduction
To drill oil wells longer than 5 km are becoming more
common now a day, being drilled from fixed platforms at a
central location in a geological basin. A new problem
accompanying long wells is the long transient time
necessary for the pump pressure or stand pipe pressure
(SPP) to reach its steady state level after a change in flow
rate has occurred. During the transient time period the
wellbore pressure is therefore unknown. A common
approach to this problem by the operator has been to wait
until steady state has been reached before resuming to the
ongoing operation.
Transient behavior has so far not been an important
problem in drilling operations. In relatively short wellbores
(< 3000 mMD), the transient period is negligible, and
therefore not a practical problem. In long wells the
accumulative transient time may become substantial. Each
change in mud pump flow rate represents a transient
situation. Frictional pressure arise with arising flow rate,
resulting in pressure increments or pressure pulses being
transmitted both downstream and upstream. The upstream
pulse is resulting from friction pressure gradually being built
up by increasing hydraulic friction further down the line.
Frictional pressure reflected from positions far away from
the flow inlet, like from bit nozzles and from the annulus, is
dissipating and distorted. It takes time for it to restore to its
true, steady state level.
Transient behavior in long wells will mask the true
characteristics of a wide range of pressure events. Such
events are wellbore kicks, breathing events, pressure testing,
down linking of pressure signals and downhole wellbore
restrictions (pack off). Wellbore restrictions tend to increase
in frequency when the well changes from short, vertical to
long, inclined path. A pack off will lead to pressure build-up
below the point of restriction. Due to transient behavior such
an event is not seen sufficiently early at the surface. In the
meantime, before it is detected, it may build up to an extent
that leads to fractures in the uncased formation below.
Rigs equipped with downhole pressure transducers are
better off, since only part of the transient behavior is taking
place in the annulus. But even with downhole recorders
installed we need to understand the physics of transient
pressure in order to interpret and counteract correctly.
2. Pressure behavior in long wells
Information on pressure transients in long wellbore was
scarce. Pressure transient behavior is in most SPE
publications related to pressure behavior in the reservoir,
described by the diffusivity equation. None of the
publications have taken the length of the wellbore itself into
consideration, although Rbearvi & Tiab [1] indicate that the
length of the horizontal wells have effect on the pressure
response during pressure drawdown tests.
One interesting and successful application of waterhammer involves overcoming mechanical friction during
coiled tubing drill CTD in horizontal wells. Mechanical
friction increases with length of wellbore. Castaneda,
Schneider and Brunskill [2] describes a tool positioned in
the bottom hole assembly (BHA), which, when a valve is
suddenly closed, creates an additional pressure pulse, its
magnitude depending on the fluid flow rate and valveclosing time.
A well-known classical water hammer problem is taking
place while shutting-in the well in on a kick (Jardine et. al.
[3]). They showed that so-called hard shut-in (meaning no
extra built-in delay) results in negligible additional pressure
below the BOP. This is explained by the long time typically
spent on closing the annular preventer followed by closing
the choke.
Figure 2: Observed transient period after pump start vs.
measured depth in the 9.5 “wellbore section. The data
point at 7 000 mMD (4 min. transient time) represents the
data set where the model was initially tested.
The aforesaid pressure transient was studied in the 9.5
inch well section. Figure 1 presents a typical example of a
transient behavior after turning the pump on and up in order
to resume the drilling operation.
Mu d
pum
SPP
p
SPP
Mud pit
Drill pipe
10:00
10:10
10:20
Annulus geometry
MFI
In this paper we will model two typical pump boundary
conditions; a) turning pump on and b) increasing pump rate
from an initial constant value. We have recorded transient
pressure behavior as a function of wellbore length. The data
are presented in Figure 2. Only the last part of the transient
period is included, the one occurring after the constant pump
rate has been reached, as indicated by the vertical line on the
time scale in Figure 1. At two selected depths the model
input geometry and rheology have been collected and
presented in Figure 3. These data will later be used as input
to the model.
unit
data
Wellbore geometry:
Drill string length
Bit size
m
in
7 000
9½
More details in Appendix
HWDP
Figure 1: Surface pressure response (SPP, in green) in a
7000 mMD long well after turning on pump mud flow
(MFI, in red) at 10:11. At 10:12:45 it is ramped up to 1750
lpm over a period of 44 seconds. After steady pump rate
is reached (at 10:13:30), marked by vertical bar) it takes
4 minutes until a steady state pressure is reached (at
10:17:30). At 10:18:00 an RPM-increase leads to an
additionally SPP increase (Statoil [4]).
Data type
Bit nozzles
Drilling fluid parameters:
n (flow index)
K
MW
Pas-n
kg/l
0.51
2.44
1.48
Steady state
perational paramters:
Pump flow rate
Pump pressure
Transient time
l/min
bar
min
1 750
177
4.0
DC
Figure 3: Principle drawing of a wellbore (left). Specific
wellbore data includes specification of geometry, drilling
fluid and some operational data.
3. The model
Transient flow is often used synonymously with waterhammer, although the latter term is customarily restricted to
water. Solving the equation of motion (momentum) and of
continuity leads to equation of pulse wave propagation,
caused by disturbances of the steady state flow. Steady flow
is a special case of unsteady flow, which the unsteady flow
must satisfy.
3.1.
Introducing the model
A simplified flow system is first used to introduce the
transient behavior. The unsteady momentum equation is
applied to a control volume of a pipe section as shown in
Figure 4, where the hydraulic friction in the pipe is
simulated by means of a valve.
A positive displacement pump is equipped with pulsation
dampeners. While running at constant rate it delivers a
constant flow velocity v0, independent of pressure. When
the flow velocity is changed by v, the change in flow
velocity is spreading downstream at the speed of a pressure
pulse, a.
(a - v0)t
pulse
rA(v0 + v)2
v0+v -a
rA(a-v0)V
+
rAgH
v0
rAv02
Figure 4: Restricting the flow in the pipe by hydraulic
friction, illustrated as a valve, results in change of
momentum in the control volume (free after Wylie and
Streeter [5]).
The velocity change v is accompanied by an additional
hydraulic friction, expressed as a head change H, which
results in a momentum change of rgH. The impulse of
the higher pressure travels upstream at the wave speed of a,
but at an absolute speed of a-v0-v. This is a reversed
sequence of what is taking place in a classical water hammer
problem in which the valve closure results in increased
head, H, accompanied by a velocity change v. However,
the drill string model and the transient effects will be
identical. Applying the momentum equation to the control
volume in Figure 4, results in equation (1), after neglecting
second order terms of v2 and v0.
H = - a  v/g

(1 + v0/a) ≈ - a  v/g
(1)
By assuming the friction valve was really a valve, a full
closure of the friction valve would result in v = - v0, and
H would become: H = a  v0/g. Latter assumption
demonstrates the classical water hammer phenomenon.
During upstart of the pump and increasing the flow rate
linearly by equal increments of v the relationship between
all velocity changes and the resulting head change becomes:
a / g   v   H
pump
 H friction  H DP  H Bit  H Ann (3)
The pulse H represents the energy related to the
hydraulic friction in the control volume. As soon as the
control volume is compressed the next one in the upstream
direction will be compressed, travelling back towards the
source, compressing the fluid, and stretching the wall a
little. When the wave reaches the upstream end, all the
upstream fluid is under the extra head, all its momentum has
been transformed to elastic energy from kinetic energy,
represented by v. The stored compressed and elastic
energy will send out a reflection at the pipe end, if not
absorbed at the discharge source.
SPP
v0 + v
 H  H
(2)
3.2.
Transient flow for viscous fluids
The method of analysis of transient flow starts with the
basic equations of motion, of continuity, state, and other
relationships of the physics involved. The selection of
restrictive assumptions dictates the overall approach. We
have selected to employ the so called characteristics method
as presented by Wylie and Streeter [5] The method converts
the two differential equations of motion and the continuity
equation into four total partial differential equations, which
are integrated to be expressed in terms of finite differences,
and finally dividing the flow system into specified time
intervals. That simplification is a challenge when pipe
sections lengths do not comply with the specific time
interval. Nevertheless, one of the many advantages of the
characteristics model is that the boundary conditions are
easily programmed.
Since a general solution to the partial differential
equations is not available, the method transforms said
equations into four total-differential equations and integrate
them to yield finite differences; easily handled numerically.
The time step results in finite length intervals, x.
Conditions of the pressure pulse are known at the two start
points A and B, and are predicted at point P, t seconds
later, through equations (4) and (5):
HP – HB + B (qP – qB) + R/4 (qB + qP)

|qB + qP| =
0
HP – HA – B (qP – qA) – R/4 (qA + qP)

|qA + qP| = 0
(5)
This is true as long as the pressure wave has not reached
the upstream end of the pipe and returned as a reflected
wave, i.e. before 2L/seconds has passed. In real cases the
reflected wave is small due to attenuation over long
distances. The minus sign is used for waves traveling
upstream.
Here B is the capacitance, B = a/g, A and R is the viscous
resistance; R = fx/(2gdA2). Eqn (4) is valid for increasing
head in the + x direction (downstream), while eqn. (5)
represents increasing head for each reach (x) in the
negative x-direction. H and q are typically known at time
zero, at which time initial steady-state conditions rule. HP
and qP are found through iteration and interpolation rules.
The steady state hydraulic friction in all the pipe section
and in singularities inside the pipe defines the steady state
HGL. At the discharge end the total friction pressure is
recorded at the stand pipe pressure gage (SPP), and
expressed through eqn. (3):
In pipeline flow the velocity is written in terms of
discharge. However, the variation of flow is unknown
during transient flow. For problems in which the friction
term dominates, a second-order approximation is necessary,
yielding equation (4) and (5).
3.3.
H0
Hydraulic grade line elevation, H
The magnitude of the pressure wave reduces due to
attenuation as it travels upwards as shown in Figure 5. At its
downstream end the friction pressure reaches its full effect
quickly. When the pressure pulse travels upstream it slows
down the initial fluid velocity, causes storage of volume as
liquid becomes compressed, and even strong pipe walls will
expands a bit. However, since pressure attenuate, and will
continue to rise for a while longer. This delayed increase is
called line packing. The pressure is approaching the steady
state HGL asymptotically.
H0
rise a
t the
w a ve
front
f =  / (½ rv2)
Distance from
discharge end
Figure 5: At t = 0 the pump is turned on and reaches
instantly a constant flow rate v0. At t = L/a seconds
later, the fluid front has reached the downstream end.
Without attenuation the final head rise H would have
been reached at the upstream end 2L/a seconds after
pump start. Dotted line represents the real, attenuated
head rise.
Combining eqn. (6) and (11 ) and solving for frictional
pressure results in the Darcy-Weisbach equation:
ppipe = f
Shear stress can be expressed in terms of pressure, the
force balance between shear stress and pressure; p /4  d2
=  dL, yielding:
dh /4L
(6)
The hydraulic diameter, dh, is equal to d for pipes and to
(douter – dinner) for annuli. Drilling fluid is non-Newtonian.
Transient flow, which is low at pump start conditions, will
experience relatively higher hydraulic friction at low flow
velocities. This behavior is taken care of through the
selected shear stress model, the Power Law model:
 n
(7)
Friction pressure depends largely on the Reynolds number,
given by
NRe = rvdh / eff
(8)
Effective viscosity of a power law fluid is given by
eff, pipe = [v/d

1/2  rv2  L/dh
(12)
(3n+1)/4n]n  Kd/8v
(13)
Eqn. (9) and (10) are slightly different from annular flow.
Shear stress is considered to be the same in transient flow as
if the velocity were steady.
Speed of a pressure pulse
In very thick-walled, stiff pipes or pipes with large
diameters, the acoustic speed of the pressure pulse is:
The hydraulic friction term
=K

Both the empirical constants are related to Power law fluid
behavior where a = (log n + 3.93) / 50 and b = (1.75 – log n)
/ 7.
3.5.

(11)
F = a  NRe-b
L
= p
(10)
For turbulent flow the friction factor must be determined
experimentally. The Fanning friction factor f is originating
from the quotient between shear stress and kinetic pressure:
Original steady state grade line
v0
3.4.
(3n+1)/4n]n  L/d
For drill pipes and drilling fluids we have selected the
Metzner & Reed (Hemeida [6]) correlation:
Final steady state HGL
Head

ppipe = 4 K [v/d
Attenuation and line packing
(9)
At laminar flow (NRe < 1 800) the frictional pressure may be
derived theoretically as:
a K/r 




In a drillpipe part of the pulse energy is absorbed by wall
streching, and a more realistic expression is (Wylie &
Streeter 1978):
a  ( K / r ) / [1  ( K / E)( D / e)]
(15)
The term c1 is adjusted for pipe being anchored at its origin
only (leading to axial streching):
c1 = 1 – /2
(16)
In the annulus c1 must instead be adjusted for cased holes:
c1 = 2Ee /( Efm D + 2Ecsg  e)
(17)
An important contribution to speed of pulse is the
volume fraction of suspended solid particles in the liquid
phase. Both K and r are adjusted:
K = Kliq /[1 + (Vs/V)(Kliq/Ks -1)]
(18)
r = rs  Vs/V + rliq  Vliq/V
(19)
4. Testing the model
4.1. Boundary conditions
The transient problem is defined by its boundary
condtions. For drilling operations the upstream-end
boundary conditions is defined by a piston pump with
known flow discharge. Head-discharge is ranging from zero
to final steady-state level. Pump characteristics behavior
during pump start are typically described by Figure 1. Flow
output can be expressed by a linearely increasing function:
q1 = q0 * t/t1, for t = 0, q = q0, for t = t1, q = q0
(20)
4.2. Model flow chart.
The program was coded in Matlab codes. Flow chart of
the program is presented in Figure 6. A tricky issue was the
determination of the specific time interval. It had to be
determined and optimized on basis of number and length of
pipe sections (see detail in flow chart).
Set values of
model constants
(mud weight, well
length, etc)
Start
Figure 7: Results of transient pressure build-up of SPP.
The data are taken from Figure 3 and Appendix.
The drill pipe has been simulated shorter and longer than
the base case in Figure 3 (and Appendix). We can state that
the model imitate the reality in an acceptable manner.
Estimated data
Discretize independent variables; Time and MD
Discretize dependent variables; q and H
Observed data
Inital conditions
for q and H
Boundary conditions
for q and H
at inlet and outlet
no
Compute grid point values
for q and H
between inlet and outlet
using equation (4) and (5)
Figure 8: Simulation of variable pipe length.
Is max time
reached?
yes
Stop
Figure 6: Flow chart of program execution.
4.3. Results
The result of the case defined in Figure 3 is presented in
Figure 7. Our results matched quite well with field
observations as shown in Figure 8.
5. Problems associated with transient pressure
Pressure inside pipes propagates at the speed of sound
waves, typically at 1 200 m/s in fluids inside steel pipes. It is
therefore common to assume that fluids accelerate
homogeneously as a stiff mass. These fast dynamics are
much faster than the bandwidth of the MPD control system
and are thus neglected in the hydraulic model (Kaasa et. al.
[7])). However, in long pipes lines (> 3 000 mMD)
including the annulus system, transient flow is exposed to
multiple reflections and to attenuation, resulting in a
transient period of minutes instead of seconds. Figure 7
demonstrates this clearly.
In Figure 9 we have selected a specific location in the
annulus to observe how the local transient pressure behaves
vs. time. We observe that some reflections are more
pronounced than others.
observed data could in this well be described by the
equation t = c  L2. The theoretical increase of transient
time vs. well length had a lower exponent than 2.
Many closed-in situations and pressure transmission
events will be exposed to long transient periods in long
wells. This can lead to safety issues and miss-functionality.
There is a need to re-visit these events to ensure full
interpretations and downhole functionality.
References
Figure 9: Transient pressure vs. time at 3 500 mMD in
the annulus. It takes 11 seconds before the first
pressure increase is seen. At 17.5 s a pronounced
reflection is coming from the outlet of the annulus and at
25 s a new one from the bit.
Pressure
In HPHT wells and other wells with narrow pressure
window it is difficult to distinguish between breathing,
kicking and transient pressure. Figure 10 illustrates how the
surface pressure will react to different events after shut-in in
long wellbores. Through proper analysis of the closure
pressure we may be able to reveal its true response.
1
2
3
4
[1]
S. A. Rbeawi and D. Tiab: Transient Pressure Analysis of
Horizontal Wells in a Multi-Boundary System. SPE paper
142316 prepared for presentation at the SPE Production and
Operations Symposium held in Oklahoma City, Oklahoma,
27-29 March 2011
[2]
J.C. Castaneda, C.E. Schneider and D. Brunskill: Coiled
Tubing Milling Operations: Successful Application of an
Innovative Variable-Water hammer Extended-Reach BHA to
Improve End Load Efficiencies of a PDM in Horizontal
Wells. SPE paper 143346, prepared for presentation at the
SPE/CoTA Coiled Tubing and Well Intervention Conference
and Exhibition held in The Woodlands, Texas, 5-6 April
2011
[3]
S.I. Jardine, A.B. Johnson, D.B. White and W. Stibbs: Hard
or Soft Shut-In: Which Is the Best Approach? SPE/IADC
paper 25712 prepared for presentation at the 1993 SPE/IADC
Drilling Conference held in Amsterdam, 23-25 February
1993
[4]
Statoil ASA: Real-time drilling data from Gullfaks. Statoil
Rotvoll, Drilling Section, Trondheim, 2007
[5]
E. B. Wylie and V. L. Streeter: Fluid transients. McGrawHill International Book Company, New York, 1978
[6]
A. M. Hemeida: Friction factor for yield less fluids in
turbulent pipe flow. J of Canadian Petroleum Technology,
Vo. 32, No 1, 1993
[7]
G.-O. Kaasa, Ø. N. Stamnes, L. Imsland and O. M. Aamo:
Simplified hydraulic model used for intelligent estimation of
downhole pressure for a managed pressure drilling control
system. SPEDC, March 2012
5
6
1. Water or oil kicks from highly permeable zone
2. Gas kick from highly permeable zone
3. Breathing wellbore
4. Kick from low permeable zone
5. Pressure transient
6. Gas rising (by bouyancy)
Time
Figure 10: Pressure responses for different pressure
events after shut-in in long wellbores.
6. Conclusion
Transient pressure is a problem only in long wellbores. In
2000 m long wells it takes typically 5-10 seconds for
pressure pulses to reach steady state, while in 7000 m long
wells it takes typically 4 minutes.
The model is based on known water-hammer theory
applied to viscous fluids. In long wellbores the transients are
caused by increased flow velocity, resulting in increased
viscous friction. The friction pressure increase is transmitted
back to the source (the pump) in an attenuated manner and
therefore taking longer time in long wellbores.
Testing of the model gave results that agreed well with
field observations. The exponential time increase of the
Nomenclature
Roman:
a
A
B
BHA
c1
d
e
E
g
H
speed of pressure pulse
cross sectional area
a/g = capacitance
bottom hole assembly
effect of pipe constraint conditions on pulse speed,
dimensionless)
diameter
wall thickness
modulus of elasticity
gravitational acceleration
pressure head
K
L
n
p
q
R
SPP
t
v
v0
V
x
bulk modulus of elasticity, consistency index in the
Power Law model
length
flow index in the Power Law model
pressure
fluid flow rate
viscous resistance
stand pipe pressure
time
fluid velocity
steady state fluid velocity
volume
axial distance
Greek:

difference, here also used for transient time

r

Poisson’s ratio
mass density
viscous stress
Subscrips:
A
0
1, 2
csg
fm
i
liq
P
s
tot
at point A
steady state condition
section numbers
casing
formation
inner; section number
liquid
predicted
solids
total