Attenuation of pump pressure in long wellbores (v02)

Journal of Petroleum Science and Engineering 122 (2014) 159–165

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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol

Attenuation of pump pressure in long wellbores (v02) Pål Skalle a,n, Tommy Toverud a,1, Stein Tore Johansen b,2 a Department of Petroleum Engineering and Applied Geophysics, NTNU, Norwegian University of Science and Technology (NTNU), S.P. Andersensveg 15a, NO-7491 Trondheim, Norway b SINTEF Materials & Chemistry/Flow Technology, Rich. Birkelandsvei 2B, NO-7465 Trondheim, Norway

art ic l e i nf o

a b s t r a c t

Article history: Received 31 May 2013 Accepted 7 July 2014 Available online 22 July 2014

Drilling of long oil wells introduces the challenge of long delay time before the pump pressure has been built up to its expected level. After pump-start in long wells the generated hydraulic friction pressure requires a transient period of several minutes, due to pressure wave attenuation, before the steady state level is reached. The true pressure along long wellbores after changing the flow rate will be unknown, since the pressure normally is recorded only at the surface. Changes in pump pressure will take place not only during pump-start, but in other operations like pressure testing, managed pressure drilling, etc. Transient pressure behavior in the field was therefore investigated. For theoretical modeling we applied the water hammer theory. Theoretically estimated transient time was compared with observed transient periods in long wellbores. Model results fitted well with observations, thus emphasizing the long transient time period. & 2014 Elsevier B.V. All rights reserved.

Keywords: oil well drilling fluid mechanics water hammer attenuation fluid flow viscous fluids pipe elasticity

1. Introduction To drill oil wells longer than 4 km is becoming more common now a days. Long wells are being drilled from offshore platforms to reach the outskirts of a geological basin. A problem related to long wells is the long pressure transient periods, i.e. the time needed to reach steady state pressure after a change in flow rate has taken place. The pressure is transient both spatially and in time. During the transient time period the true wellbore pressure is therefore more or less unknown along the well, since it is normally measured only at the surface. The field approach to meet this problem has been to wait until steady state has been reached before resuming the ongoing drilling operation. Attenuated pump pressure has so far not been an important problem in drilling operations. In relatively short wellbores (o4000 mMD), the delay period is negligible (seconds), and therefore not a practical problem. In longer wells (44000 mMD) the accumulative delay time may become substantial (minutes). Each change in the mud pump flow rate represents a transient situation. Frictional pressure increases when the flow rate is increased, is a direct response to the downstream friction pressure. The information of the downstream pressure increase must be transmitted back to the discharge pump.

n

Corresponding author. Tel.: þ 47 918 97 303. E-mail addresses: [email protected] (P. Skalle), [email protected] (T. Toverud), [email protected] (S.T. Johansen). 1 Fax: þ47 73 94 44 72. 2 Fax: þ47 73 59 70 43. http://dx.doi.org/10.1016/j.petrol.2014.07.005 0920-4105/& 2014 Elsevier B.V. All rights reserved.

Reported studies on pressure transients related to the length of the pipe in the petroleum industry were scarce. Studies on pressure transient behavior have mostly been related to two areas; to well testing during production and to surveillance of transport of oil and gas in pipelines. The majority of the publications were focusing on gas production. Surveillance and diagnosis of long pipelines are vital and need an accurate transient model for operations like detection of pipeline leakage, well shut-in and production startup/restart. Ling et al. (2012) and Adeleke et al. (2012) are two representatives of such investigations. They investigated surface pressure response after sudden restriction occurring downstream through 1D partial differential equation based on conservation of mass and momentum. Reported two-phase flow transient period was largely dictated by the gas compressibility, and none of the studies considered the effect of pipe length. The second group of transient pressure studies was well testing (e.g. Raba and Ertekin, 2012), also called draw-down tests. The reservoir responds with useful transient data. The transient data are compared to type curves to determine the permeability and the extent of artificially induced fractures. Type curves, however, do not contain the transient behavior related to the length of the wellbore, although Rbeawi and Tiab (2011) indicated that the length of the horizontal wells have an effect on the pressure response during pressure drawdown tests, but without quantifying it. Pierre and Gudmundsson (2011) developed a model for one phase transient oil flow. Their model of transients, based on conservation of mass and momentum, fitted well with observations. This study involved a sudden local flow change in a

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Nomenclature

Greek

a a A C c1

Δ μ

d, D e E f g H HGL K K L n p q R SPP t v v0 V x

speed of pressure pulse, m/s acceleration, m/s2 cross sectional area, m2 pipe capacitance, g A/a2, m s2 effect of pipe constraint conditions on pulse speed, dimensionless diameter, m wall thickness, m Young's modulus of elasticity of material, Pa Fanning friction factor, dimensionless gravitational acceleration, m/s2 pressure head, m hydraulic grade line, m bulk modulus of elasticity for the fluid, Pa consistency index in the Power Law model, Pa s  n length, m flow index in the Power Law model, dimensionless pressure, Pa fluid flow rate, m3/s viscous resistance, dimensionless stand pipe pressure, Pa time, s fluid velocity, m/s steady state fluid velocity, m/s volume, m3 axial distance, m

relatively short (2.7 km) pipeline. They pointed out that the material's elastic properties in pipelines, consisting of several concentric layers, were both inaccurately modeled and specified. Due to the lack of awareness of the transient pressure phenomenon in long wellbores, the influence of wellbore length on pressure transients was therefore investigated by present authors. From several wellbores, a 9.5 in. well section was selected for demonstrating transient behavior. Fig. 1 presents the base–case well and its geometry. Fig. 2 presents a typical example of a transient behavior after turning the pump on. In that well we recorded transient pressure behavior as a function of wellbore length. The observed data are presented in Fig. 3. Only the end part of the transient period is included, the one occurring after the pump rate increase was completed, as indicated by the vertical line on the time scale in Fig. 2. Curve fitting of the data plot in Fig. 3 showed that transient time ttransient, increased exponentially with wellbore length L (with an associated regression constant r2 ¼0.98) t transient ¼ cL1:91

ð1Þ

The theoretical approach had already been derived under the heading of water hammer. The solution technique which fitted well for our purposes was derived by Wylie and Streeter (1978). We adopted their solution technique and could apply it more or less directly. We will therefore first present the basic ideas behind the water hammer phenomenon and then the more complex solution technique for viscous fluids; the so-called Characteristic Method.

2. Classical water hammer theory Transient flow is often used synonymously with water–hammer, although the latter term is customarily restricted to water, as

ρ τ

difference, dimensionless Poisson's ratio of deformation dimensionless mass density, kg/m3 viscous stress, Pa

of

material,

Subscrips A, B 0 1, 2 csg fm i liq P s tot

at points A, B steady state condition section numbers casing formation inner; section number liquid predicted solids total

Metric conversion factors in bar

0.0254 m 105 Pa

the name suggests. In the present paper we will apply Wylie and Streeter's (1978) model of water hammer on transient pressure signals in oil wells. In oil wells the theory will not be applied for shutting down the flow, which is the common case of water hammer, but rather for increasing the flow. The result of these two opposite actions is similar; a transient pressure is generated. We intend to make the readers aware of the potentially dangerous phenomena and associated problems. The main assumptions behind the model are summarized here (1) To accommodate the pressure–wave speed the fluid is assumed isentropic compressible: ρ ¼ ρ0 þ∂ρ/∂p|S (p p0) and sound speed a¼ (1/∂ρ/∂p|S)0.5; (2) The fluid's density is only mildly influenced by pressure. This effect on the mean flow is therefore neglected; (3) The fluid is for other purposes assumed incompressible, and only the wall's elasticity is impacting (dampening) the wave speed; (4) The wave pressure amplitude is dampened by viscous shear (dissipation); (5) The pressure wave is partially reflected at every area change and partly transmitted further downstream, while fully reflected at channel ends. Reflections involve phase shift; (6) The two-phase system is assumed to be a liquid phase with continuous dispersed solids with no complex wave speed effects. In classical water hammer theory, the dependent variable pressure p is converted to pressure head H, also referred to as piezometric head, the elevation above an arbitrary datum; p ¼ ρgH

ð2Þ

All factors involved in the momentum changes in a control volume when closing a valve are summarized in Fig. 4. The figure is modified to fit the case when a pump is turned on and up, rather than abruptly closing a valve. The valve represents hydraulic pipe friction vs. pipe length. While initially running the mud pump at constant rate, it delivers a constant flow velocity of v0. By increasing the flow rate by the upstream located pump (a positive

P. Skalle et al. / Journal of Petroleum Science and Engineering 122 (2014) 159–165

Mud

pum

161

SPP

p

Mud pit

Drill pipe

Annulus geometry HWDP

Bit nozzles

Data type

unit

data

Wellbore geometry: Drill string length Bit size

m in

7 000 9½

Drilling fluid parameters: n (flow index) K Mud density

Pas-n kg/l

0.51 2.44 1.48

Steady state operational parameters: Pump flow rate Pump pressure Time to reach constant pressure

l/min bar min

1 750 177 4.0

DC

Fig. 1. Principle drawing of a wellbore (left). Specific wellbore data includes specification of geometry, drilling fluid and operational data for testing the model.

P

pre ump

ssur

e

300

3000 Flow rate

200

100

0 10:00

p

4000

2000

1000

Flow rate (lpm)

Pump pressure (bar)

400

(a - v0)Δt

pulse

v0 + Δ v

0 10:10

10:20

Fig. 2. Surface pump pressure response in a 7000 mMD long well after turning on the pump mud flow, first at 10:11. At 10:12:45 it is ramped up to 1750 lpm over a period of 44 s. After steady pump rate is reached (at 10:13:30), marked by the vertical bar, it takes 4 min until a steady state pressure is reached (at 10:17:30). Steady state is defined as pressure variation less than þ /  2 bars over three time steps (15 s). At 10:18:00 an RPM-increase leads to an additionally pump pressure increase. The data were provided by Statoil (2007).

ρA(v0 + Δv)2

v0+Δv -a

ρA(a-v0)Δv + ρAgΔ Η

v0

ρAv02

Fig. 4. A circulation system during horizontal drilling represented by a control volume in the drill string. Restricting the drill string, here illustrated by a valve, results in change of momentum in the control volume, presented in the bottom line (free after Wylie and Streeter (1978)).

A full closure of the valve would mean that Δv is equal to – v0, and ΔH would accordingly become equal to av0/g, which demonstrates the classical water hammer phenomenon. During upstart of the pump by increasing the flow rate linearly through equal increments of Δv the relationship between all velocity changes and the resulting head change becomes  ∑ΔH ¼ a=g∑Δv

ð4Þ

In very thick-walled, stiff pipes, the acoustic velocity of a pressure pulse is a function of the fluid's bulk modulus K and its density ρ Fig. 3. Observed transient time after pump start vs. measured length in a 9.5 in. wellbore section.

displacement pump) the velocity increases by Δv. The velocity change Δv is accompanied by an additional hydraulic friction, expressed as head change ΔH, which results in the momentum change of ρΑgΔH. A resulting pressure pulse is reflected back upstream at its wave speed, but slightly reduced by the steady fluid velocity; v0. After neglecting second order terms of Δv and v0, the momentum equation, presented in Fig. 4, reduces to

ΔH ¼  aΔv=gð1 þ v0 =aÞ   aΔv=g

ð3Þ

a ¼ √ðK=ρÞ

ð5Þ

See Appendix for adjustments in acoustic velocity vs. pipe elasticity and fluid composition. Here Eq. (A3) shows that the wave speed is proportional to the modulus of elasticity of the wall; the lower the modulus (softer material) the lower the wave speed.

3. Transient flow of viscous fluids Transient flow in viscous fluids is more complex than for water. Wylie and Streeter (1978) suggested converting the differential

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P. Skalle et al. / Journal of Petroleum Science and Engineering 122 (2014) 159–165

equations of motion and of continuity into four total partial differential equations, integrating and expressing them in terms of finite differences, and finally applying the characteristics method to solve them. The characteristics method is a useful approach with several advantages. One of the many advantages was that the boundary conditions were easily programmed since finite differences are easily handled numerically. In the present paper we present the two basic equations and the resulting solution method. First the equation of motion: here four different forces act on a free fluid body of cross section area A; the pressure forces acting normal to the flow direction; the shear stress components acting on the periphery faces opposing the flow direction x. The third force is gravity. The flow pipe is inclined with the vertical at an angle I, positive when the elevation decreases in the þx direction. A ∂p=∂x  π dτ þ ρgA cos I ¼ Aρ Dv=Dt

ð6Þ

In sum they are equal to the fluid's mass times acceleration, the total derivative. The continuity equation states that the time rate of mass inflow into a control volume is exactly equal to the time rate of change of mass within the control volume: ∂ðρAvÞ=∂x þ ∂ðρAÞ=∂t ¼ 0

ð7Þ

Eq. (7) incorporates volume changes, both of the fluid and of the pipe.

leading to HP  HB þ

a 2f Δx 2 ðq  qB Þ þ qB  0 gA P DgA2

ð9Þ

In Eq. (9) the viscous resistance R is equal to 2fΔx/(2gdA2). In the Appendix an expression of the Moody friction factor f is presented for non-Newtonian fluids. Eq. (9) is valid for increasing pressure head. A corresponding equation is needed for decreasing pressure (reflecting pressure wave). H and q are known at time zero, at which time the initial steady-state conditions are ruling at the start point A (and B), and are predicted at point P, Δt seconds later. For details of transforming the basic equations please refer to Wylie and Streeter (1978), and to the Flow chart of the program, presented in Fig. 6. A finite difference step-wise scheme was used for the numerical modeling. Smaller step lengths in the finite difference modeling yield better results, but this comes at the cost of increased computation time. The spatial and temporal step lengths are connected through the characteristic equations. The length of the pipe sections would preferably correspond to integer multiples of the spatial step length, but to comply with this requirement would be too computationally costly due to the small step lengths this would require. Instead, compliance is achieved by small adjustments to the pipe section lengths. These adjustments are assumed to introduce only small errors.

5. Results 4. The developed model for solving transient behavior and its solution procedure One of the forms of the equation of motion, valid for increasing pressure head, is expressed through Eq. (8). Pressure head H and fluid velocity v are the two dependent variables, both being functions of x and t; through dx/dt ¼ a

To test the model we applied the wellbore defined in Fig. 1. Geometry and rheology were collected at four selected depths. The results of applying the solution procedure on the base case and the additional drill sting lengths are presented in Fig. 7. The total transient time is determined when the normalized head reached 1.0.

dp dv 4 þ ρa þ τ  0; dx dx D which can be written as g

dH dv 4 dH dv 2 þ a þ τ  0-g þ a þ f v2  0 dx dx ρD dx dx D

ð8Þ

By expressing velocity in terms of discharge rate q, the two basic equations can be made suitable for integration along the characteristic lines, expressed through Eq. (9) and in Fig. 5. From g

dH dv 2 2 Δx 2 þ a þ f v2  0 ) g ΔH þaΔv þ fv 0 dx dx D D

yielding

ΔH þ

a 2Δx  q 2 f Δq þ 0 gA Dg A

t

P

-

+

Δt A

1 i+

Δx

Δx i

i-1

Δx

B

x

Fig. 5. Characteristic lines in the x–t plane. Position is related to time through the wave propagation velocity (free after Wylie and Streeter (1978)).

Fig. 6. Flow chart of program execution.

P. Skalle et al. / Journal of Petroleum Science and Engineering 122 (2014) 159–165

163

Fig. 7. Results of transient pressure buildup. The well data are taken from Fig. 1. The left-hand plot represents two medium long wells, and to the right a plot of four long wells.

Fig. 9. Simulation of variable pipe length seen together with observed data.

Fig. 8. Transient pressure vs. time at 3500 mMD in the annulus section in a 7000 mMD long well. It takes 11 s before the very first pressure increase is seen here. At 17.5 s a pronounced reflection is coming from the rock bit and at 25 s a secondary reflection is noted. In 35 s the pressure here has reached 87% of its final stationary pressure.

In short drill strings oscillation are transmitted to the surface where several reflections are observed. Multiple reflections are damped in the viscous fluid, and finally they fade out. In drill strings somewhat longer than 2200 mMD, even the largest (the first one) oscillation had vanished. In Fig. 8 we have selected an observation location in the annulus midway between the bottom and the surface, i.e. at 3500 mMD, to observe how the local transient pressure behaves vs. time here. We saw traces of reflections/oscillations, and we also observed that at this position the signal was less attenuated than at the discharge end. When the simulated results are plotted together with field observations as shown in Fig. 9, it is seen that our results matched sufficiently well to state that there is agreement between model and field observations. The important point of long delays of the pressure signal has been proven.

6. Explanation of the long delay time The long delay time of reaching the steady state pressure is related to attenuation or line packing. The magnitude of the

Fig. 10. At t¼ 0 the pump rate is increased from v0 and reaches instantly a constant velocity of v0 þ Δv. At Δt ¼L/a seconds later, the fluid front has reached the downstream end. Without attenuation the first surface head rise ΔH would have been reached at the upstream end 2 L/a ( ¼ 2  7000/1400¼ 10 s) seconds after pump start. Dotted line represents the real, attenuated head rise (free after Wylie and Streeter (1978)).

pressure attenuation as it travels upwards towards the source is represented in Fig. 10. When fluid motion is changed by the pump, the front of the increased motion is first transmitted downstream by the speed of the compression wave. At any distance downstream, the increased mass flow will be exposed to increased hydraulic restriction, resulting in a new compression of the fluid, but with opposite direction than the first. Latter compression pulse is transmitted upstream until all the fluid has been compressed back to the source; the pump pressure has been incrementally increased. At the same time the downstream flowing fluid is momentarily slightly slowed down by the opposing pulse and by the elastic pipe which will stretch and expand. As liquid becomes compressed storage of volume will take place. Some of the fluid's kinetic energy has in this way been converted to elastic energy. At

6 km 9 km

Downhole, un-delayed response Delayed surface response in long wells

P tu um o f rne p f d

0 km 3 km

Pressure response

P. Skalle et al. / Journal of Petroleum Science and Engineering 122 (2014) 159–165

Pressure response

164

Surface and downhole pack-off response in short wells

t

t

Fig. 11. Pressure responses in short and long wells. Left: a surface pressure signal is sent out and changed when reaching the bottom due to attenuation. Right: a downhole packoff is detected in short wells by the driller who reacts by turning down the mud pump. In long well the packoff is un-detected for a long time (see dotted lines). The packoff pressure keeps building until it either breaks up the pack or develops until the pressure exceeds the fracture pressure.

the source end the incremental increased fluid pressure is resulting in a new incremental pressure pulse which is sent downstream. The progressively repeated pressure increase both as a function of increased distance and as a function of time (cyclic interchange of information of pressure increase) in an iterative manner until the friction pressure in the complete pipe system has asymptomatically reached its final steady state friction pressure, corresponding to the level of flow rate.

7. Practical problems associated with transient pressure Pressure inside pipes propagates at the speed of sound waves, typically at 800–1200 m/s in fluid-filled, steel pipes. It is common to assume that a fluid accelerate homogeneously as a stiff mass, and thus transmits pressure instantaneously. This is true only in short pipes. Kaasa et al. (2012) assumed that the pulse dynamics are much faster than the bandwidth of the MPD control system. However, in long pipe lines (4 4000 mMD) long transient time will ruin this assumption, and rethinking and redesign seems necessary. 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 draw-down testing, and down-linking of pressure signals. As indicated in Fig. 11 a surface generated pressure signal is completely different from the local, downhole pressure. Downhole wellbore restrictions (packoff) tend to increase in frequency when the well path changes from short and vertical to long and inclined. A packoff will lead to pressure buildup below the point of restriction as presented in Fig. 11. Due to transient behavior such an event is not seen sufficiently early at the surface. In the meantime, before the packoff is detected at the surface, it may build up to such an extent that the uncased formation below the packoff position may fracture. Rigs equipped with downhole pressure transducers are better off, since the distance from the point of action is shorter. But even with downhole recorders installed we need to understand the physics of transient pressure in order to interpret and counteract it correctly.

9. Conclusion Transient pressure during drilling operations is a problem only in long wellbores. In a 1000 m long wellbore it takes typically 5–10 s for pressure pulses to reach steady state, while in 7 km long wells it takes typically 4 min. It is a challenge to understand the physics of pressure transients in a complex flow system like the wellbore and its annulus, involving singularity pressure losses through the BHA, and line packing caused by continuously increasing pressure resistance with increasing flow length. The model applied in this study was based on known waterhammer theory applied to viscous fluids. The investigated pressure transients were created by increased mud pump flow rate, resulting in increased hydraulic friction. The incremental friction pressureincrease/line packing was transmitted back to the fluid source (the pump) in a cyclic manner. It took long time before stationary pressure was reached. Transient pressure behavior is governed by two effects: (1) the wave speed itself is slowed down by elastic channel walls and (2) progressive pressure increase is reflected to the mud pump from downstream reflectors. The observed increase in transient time can be described by the relationship ttransient ¼cL0.91. Testing of the theoretical model gave results that agreed well with field observations. Many closed-in situations and pressure transient events are exposed to long transient periods in long wells. In addition to mud pump flow rate increase, two examples were presented, well testing and packoff situations, in which situations we depend on knowledge of the pressure transient itself. In such examples it is vital to know the signature of the transient pressure; otherwise it can lead to misinterpretations and even to safety issues.

Acknowledgment NTNU is acknowledged for supporting us during research and for encouraging Open Access publications.

Appendix A A1: adjusted acoustic velocity

8. Further work After having pointed out the problem and tested the physics through a simple but efficient model, we plan to improve the model. Elastic wall behavior needs to be explicitly incorporated on our in-house model. One question is the effect of the velocity profile on wall shear stress, especially at low flow rate. Another uncertainty is the cuttings' effect on wave speed.

Part of the pulse energy in a drillpipe is absorbed by the expanding walls, and a more realistic expression of wave speed is (Wylie and Streeter, 1978)
a ¼ fðK=ρÞ=½1 þ K=E ðD=eÞc1 g0:5 ðA1Þ The term c1 is adjusted for pipe being anchored at its origin only (leading to axial stretching): Table A1. c1 ¼ 1–μ=2

ðA2Þ

P. Skalle et al. / Journal of Petroleum Science and Engineering 122 (2014) 159–165

Table A1 Drill string and BHA. When longer string than 5000 m a 65/8 in. (ID ¼5 in.) was applied. Parameter Unit

OD (in.)

ID (in.)

Length (m)

Akkumulative length (m)

Bit þ bit sub Stab Collar Stab HW drill pipe Drill pipe Drill pipe

9.5 9.4 6.5 9.3 5.000 5.000 5.9

3.0 2.4 3.0 3.0 3.3 4.7

1.6 2.4 20 3 123 2000 2850

1.6 4 24 27 150 2150 5000

f ¼ aN R b

c1 ¼ 2Ee=ðEf m D þ 2Ecsg eÞ

ðA3Þ

An important contribution to the pulse velocity is the volume fraction of suspended solid particles in the liquid phase. Both K and ρ must be adjusted K ¼ K liq =½1 þ ðV s =VÞðK liq =K s 1Þ

ðA4Þ

ρ ¼ ρs V s =V þ ρliq V liq =V

ðA5Þ

A2. viscous resistance for non-Newtonian fluids: Viscous shear resistance is derived from the force balance between shear stress and pressure; Δpπ⧸4d2 ¼ τπdL. Shear stress τ can be expressed in terms of pressure

τ ¼ Δpdh =4L

ðA6Þ

The hydraulic diameter, dh, is equal to d for pipes and to (douter–dinner) for annuli. Drilling fluids are normally nonNewtonian. Transient flow rate, 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:

τ ¼ K γn

ðA8Þ

At laminar flow (NRe o 1800) the frictional pressure for pipe flow is expressed as

Δppipe ¼ 4K½8v=dð3n þ 1Þ=4nn L=d

References Adeleke, N., Ityokumbul, M.T., Adewumi, M., 2012. Blockage detection and characterization in natural gas pipelines by transient pressure-wave reflection analysis. SPE J. April, 355–365. Hemeida, A.M., 1993. Friction factor for yield less fluids in turbulent pipe flow. J. Can. Pet. Technol. 32 (1) o http://hdl.handle.net/123456789/5789 4. Kaasa, G.-O., Stamnes, Ø.N., Imsland, L., Aamo O.M., 2012. Simplified Hydraulic Model used for Intelligent Estimation of Downhole Pressure for a Managed Pressure Drilling Control System. SPEDC, March. Ling, K., Guo, B., Zhang, H., 2012. Numerical simulation of transient flow in a gas pipeline and tank. Oil Gas Facil. December, 46–56. Pierre, B., Gudmundsson, J.S., 2011. Water hammer in offloading systems: contribution of detailed modeling to design and safety. SPE Projects, Facil. Constr. September, 132–137. Raba, S., Ertekin, T., 2012. Type Curves for Pressure Transient Analysis of Composite Double-Porosity Gas Reservoirs. SPE Paper 153 889, Presented at the SPE Western Regional Meeting, Bakersfield, 19–23 March. Rbeawi, S.A., Tiab, D., 2011. 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. Statoil, A.S.A., 2007. Real-Time Drilling Data from Gullfaks. Statoil Rotvoll, Drilling Section, Trondheim. Wylie, E.B., Streeter, V.L., 1978. Fluid Transients. McGraw-Hill International Book Company, New York.

Pål Skalle is an Associate Professor at the Norwegian University of Science and Technology (NTNU), Norway. He received his Ph.D. in 1983 in Drilling Engineering from NTNU. He has served both as Technical and Associate Editor for the journal SPED&C, and was in 2005 honored by SPE as an “Outstanding Technical Editor” and in 2008 as “A Peer Apart”. Skalle's present research activities and fields of interest covers topics like Knowledge/Ontology Engineering, Drilling Fluid Engineering, Evaluation of Real Time Drilling Data and Pressure control during drilling.

ðA9Þ

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: f ¼ τ=ð1=2ρv2 Þ

ðA14Þ

ðA7Þ

Friction pressure depends largely on the Reynolds number, given by N Re ¼ ρvdh =μef f

ðA13Þ

The empirical constants a and b are related to Power law fluid behavior; a¼ (log nþ 3.93)/50 and b ¼(1.75–log n)/7. This correlation tends to overestimate the frictional pressure losses, but its simplicity compensate for an acceptable loss of accuracy. Shear stress is considered to be the same for transient flow as for steady flow. The steady state hydraulic friction in all pipe sections and in all singularities in the pipe and the annulus define the steady state friction pressure. At the inlet end the total friction pressure is recorded as the pump pressure and expressed as H pump ¼ ΔH f riction ¼ ΔH DP þ ΔH BHA þ ΔH Ann

For annular conditions c1 must likewise be adjusted for cased and cemented wellbore walls:

165

ðA10Þ

Tommy Toverud is an associated lecturer at the Norwegian University of Science and Technology (NTNU), Norway. He received his Ph.D. in 2003 in Petroleum Geophysics from NTNU. His main interest is pressure signals and their interference with the surroundings in geophysical or engineering processes, where he has authored and co-authored several papers.

By combining Eqs. (A6) and (A10) the Darcy-Weisbach equation is obtained:

Δp ¼ f ð1=2Þρv2 ðL=dh Þ

ðA11Þ

For pipe flow the dimensionless friction factor is defined for laminar by Eq. (A12) and for turbulent flow through the Metzner and Reed (Hemeida, 1993) correlation in Eq. (A13): f ¼ 16=N R

ðA12Þ

Stein Tore is a Principal Scientist at SINTEF Materials. In 1990 he was awarded the degree of Dr. Tech. based on the work "On the modeling of disperse two-phase flows", 1990. Since 1993 Johansen has been an Adjunct Professor at Department of Energy and Process Technology, NTNU, and published several text books and numerous papers within material transport.