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`Copyright
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`by
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`Somnath Mondal
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`2010
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`IWS EXHIBIT 1060
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`EX_1060_001
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` The Thesis Committee for Somnath Mondal
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` Certifies that this is the approved version of the following thesis:
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` Pressure Transients in Wellbores: Water Hammer Effects and
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`Implications for Fracture Diagnostics
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` APPROVED BY
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` SUPERVISING COMMITTEE:
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`Supervisor:
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`Mukul M. Sharma
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`David DiCarlo
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`IWS EXHIBIT 1060
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`EX_1060_002
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`

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`
` Pressure Transients in Wellbores: Water Hammer Effects and
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`Implications for Fracture Diagnostics
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`
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`
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`by
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` Somnath Mondal, B.E.
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` Thesis
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`Presented to the Faculty of the Graduate School of
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`The University of Texas at Austin
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`in Partial Fulfillment
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`of the Requirements
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`for the Degree of
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` Master of Science in Engineering
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` The University of Texas at Austin
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` December, 2010
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`IWS EXHIBIT 1060
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`EX_1060_003
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`

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`Dedication
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`To my family and friends.
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`IWS EXHIBIT 1060
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`EX_1060_004
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`Acknowledgements
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`It is a pleasure to thank those who made this thesis possible. I owe my deepest
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`gratitude to my supervisor, Dr. Mukul M. Sharma, for his constant encouragement,
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`guidance and support over the past two years. I would like to thank Dr. David DiCarlo for
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`taking time to read and review this work.
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`I would like to thank the members of the Hydraulic Fracturing and Sand Control
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`Joint Industry Project (JIP) at the University of Texas for their financial support. I would
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`like to specifically thank Mr. Adi Venkitaraman of Chevron for providing field data. I
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`also express my sincere gratitude to Dr. George Wong of Shell for his expertise that has
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`been immensely helpful in formulating this problem, and also for providing field data.
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`I would like to acknowledge Jin Lee for her support in maintaining this wonderful
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`research group. I thank my family and friends for making me who I am.
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`December, 2010
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`v
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`IWS EXHIBIT 1060
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`EX_1060_005
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`Abstract
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`
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` Pressure Transients in Wellbores: Water Hammer Effects and
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`Implications for Fracture Diagnostics
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`
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`
`
`Somnath Mondal, MSE
`
`The University of Texas at Austin, 2010
`
`
`
`Supervisor: Mukul M. Sharma
`
`
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`A pressure transient is generated when a sudden change in injection rate occurs
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`due to a valve closure or injector shutdown. This pressure transient, referred to as a water
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`hammer, travels down the wellbore, is reflected back and induces a series of pressure
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`pulses on the sand face. This study presents a semi-analytical model to simulate the
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`magnitude, frequency and duration of water hammer in wellbores. An impedance model
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`has been suggested that can describe the interface, between the wellbore and the
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`formation. Pressure transients measured in five wells in an offshore field are history
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`matched to validate the model. It is shown that the amplitude of the pressure waves may
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`be up to an order of magnitude smaller at the sand face when compared with surface
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`measurements. Finally, a model has been proposed to estimate fracture dimensions from
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`water hammer data.
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`Table of Contents
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`List of Tables ......................................................................................................... ix
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`List of Figures ..........................................................................................................x
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`Chapter 1: Introduction ............................................................................................1
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`1.1 Literature Review......................................................................................5
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`1.1.1 Water hammer Modeling ..............................................................5
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`1.1.2 Fracture Impedance .......................................................................8
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`Chapter 2: Model Formulation...............................................................................10
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`2.1 Water hammer Modeling Equations .......................................................10
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`2.1.1 Continuity Equation ....................................................................10
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`2.1.2 Equation of Motion .....................................................................11
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`2.1.3 Velocity of Water Hammer Waves .............................................12
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`2.2 Method of Characteristics .......................................................................12
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`2.2.1 Finite Difference Equations ........................................................16
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`2.2.2 Nomenclature ..............................................................................18
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`2.3 Boundary Conditions ..............................................................................19
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`2.3.1 Flowrate as a Specified Function of Time at Upstream End ......20
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`2.3.2 Series Connection .......................................................................20
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`2.3.3 Downstream Boundary Condition ..............................................21
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`2.3.4 Definition of Hydraulic Impedances ...........................................23
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`2.3.5 Analogous Electrical Circuit Representation ..............................25
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`2.4 Fracture Impedance .................................................................................30
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`2.4.1 Fracture Dimensions from Model Parameters ............................30
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`2.4.2 Estimate of Model Parameters ....................................................33
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`Chapter 3: Results and Discussion .........................................................................36
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`3.1 Hydraulic Impedance Testing .................................................................36
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`3.2 History Matching Surface Water Hammer in Injectors ..........................38
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`3.3 Simulated Bottomhole Water Hammer in Injectors ................................38
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`3.4 Fracture Diagnostics in Injectors ............................................................51
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`3.5 Fracture Diagnostics from Minifrac Data ...............................................51
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`Chapter 4: Conclusion............................................................................................54
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`Appendix ................................................................................................................56
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`References ..............................................................................................................57
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`List of Tables
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`Table 3.1: Summary of model parameters, total surface and bottomhole pressure
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`fluxes and attenuation. ......................................................................49
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`Table 3.2: Summary of model parameters, equivalent fracture dimensions and near
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`wellbore frictional pressure drop. .....................................................49
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`Table 3.3: Comparison of fracture dimensions obtained from model with dimensions
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`from fracture simulator for a minifrac job. .......................................52
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`List of Figures
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`Figure 1.1: Conceptual schematic of water hammer in a reservoir-pipe-valve system
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`(From KSB Know-how series on Water Hammer) .............................4
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`Figure 2.1: Characteristic grid in the x-t plane. .....................................................15
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`Figure 2.2: Characteristic lines in the x-t plane. ....................................................16
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`Figure 2.3: Nomenclature scheme for water hammer analysis. .............................19
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`Figure 2.4: Valve opening and closing as a function of time. ...............................20
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`Figure 2.5: Schematic of a minifrac connected to a wellbore................................26
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`Figure 2.6: Electrical circuit representation of a minifrac. ....................................27
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`Figure 2.7: Schematic of an injection well. ...........................................................28
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`Figure 2.8: Electrical circuit representation of an injector. ...................................28
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`Figure 2.9: Schematic of water hammer decline and near wellbore average pressure.
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`...........................................................................................................30
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`Figure 3.1: Simulated HIT for well with open fracture. ........................................37
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`Figure 3.2: Simulated HIT for well with closed fracture. ......................................37
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`Figure 3.3: History matching overall surface water hammer in Well A. ...............41
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`Figure 3.4: Detailed waveform comparison of water hammer in Well A. ............41
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`Figure 3.5: History matching overall surface water hammer in Well B. ...............42
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`Figure 3.6: Detailed waveform comparison of water hammer in Well B. .............42
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`Figure 3.7: History matching overall surface water hammer in Well C. ...............43
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`Figure 3.8: Detailed waveform comparison of water hammer in Well C. .............43
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`Figure 3.9: History matching overall surface water hammer in Well D. ...............44
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`Figure 3.10: Detailed waveform comparison of water hammer in Well D. ..........44
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`Figure 3.11: History matching overall surface water hammer in Well E. .............45
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`Figure 3.12: Detailed waveform comparison of water hammer in Well E. ...........45
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`Figure 3.13: Misrepresentation of water hammer data due to the effect of under-
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`sampling (after Wang et al., 2008)....................................................46
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`Figure 3.14: Simulated bottomhole water hammer for well A ..............................46
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`Figure 3.15: Simulated bottomhole water hammer for well B. .............................47
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`Figure 3.16: Simulated bottomhole water hammer for well C. .............................47
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`Figure 3.17: Simulated bottomhole water hammer for well D. .............................48
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`Figure 3.18: Simulated bottomhole water hammer for well E. ..............................48
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`Figure 3.19: Simulated bottomhole water hammer for a downhole shut-in in well A.
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`...........................................................................................................50
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`Figure 3.20: Simulated bottomhole water hammer for a downhole shut-in in well A
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`with different formation properties. ..................................................50
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`Figure 3.21: Comparison of modeled and measured surface water hammer pressure
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`for a minifrac job. .............................................................................53
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`Figure3.22: Comparison of modeled and measured bottomhole water hammer data for
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`a minifrac job. ...................................................................................53
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`Chapter 1: Introduction
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`A change in flow causes a change in pressure, and vice-versa, which leads to
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`transients in hydraulic systems. Water hammer is a surge or pressure wave that is created
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`due to a sudden change in flow velocity in a confined system. It is a transient
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`phenomenon that may be triggered by abrupt opening or closing of valves, starting or
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`stopping of pumps, failure of mechanical devices in a flow line, etc. The name, water
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`hammer, originates from the hammering sound that sometimes accompanies this
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`phenomenon (Parmakian, 1963). The variation in pressure due to water hammer can be
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`large, sometimes in the order of thousands of psi. The pressure fluctuations then
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`propagate in the system like a wave and may cause severe damage.
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`A conceptual schematic of water hammer in a simple system is shown in Fig. 1.1.
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`The system consists of a frictionless horizontal pipe of constant diameter, which is fed by
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`a reservoir at constant pressure, and is connected to a downstream valve that is suddenly
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`closed.
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`1. At
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`, the pressure head is steady down the length of the pipe, as
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`shown by the constant hydraulic grade line (shown in red), because
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`friction was neglected, and the flow velocity is
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`.
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`2. As soon as the valve is shut-in, the fluid element closest to the valve
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`comes to rest, and this rate of change of momentum causes a rise in the
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`pressure head by
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`. As subsequent fluid elements come to rest, the
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`high pressure propagates upstream from the valve towards the reservoir
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`like a pressure wave.
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`3. At
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`, where L is the pipe length and a is the wave speed, the high-
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`pressure wave reaches the reservoir as all the fluid in the pipe comes to
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`
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`1
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`t  0
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`v 0
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`  H
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`t  L a
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`rest. However, this causes a pressure discontinuity at the boundary with
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`the constant pressure reservoir.
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`4. In order to achieve pressure equilibrium at the reservoir, a pressure wave
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`of magnitude
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` is reflected back towards the valve and the direction
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`of the flow velocity reverses towards the reservoir. This reflected wave
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`reaches the downstream valve at
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`. This time is called the
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`reflection time,
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`.
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`5. At
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`, the flow velocity in the entire pipe is
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`. This causes
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`another discontinuity at the downstream valve, where the velocity must
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`be zero.
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`6. The change in velocity from
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`to zero, cause a sudden negative change
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`in pressure of
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`. This low-pressure wave travels upward as the fluid
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`in the pipe again comes to rest, reaching the reservoir at
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`.
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`7. At
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`, the fluid in the pipe is at rest but there is a discontinuity at
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`the constant pressure reservoir boundary.
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`8. As the pressure resumes the reservoir pressure, a wave of increased
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`pressure originating from the reservoir travels back to the valve as the
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`flow velocity in the pipe changes to
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`.
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`9. At
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`, the conditions in the system are the same as 1, and the whole
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`process starts over again.
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`Some of the earliest studies and experiments in water hammer were done by
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`Joukowsky (1900). The Joukowsky equation states that the rise in peizometric head
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`(
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`) due to the fast shut-in of a downstream valve (
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`) is given by:
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`2
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`  H
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`t  2 L a
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`T r
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`t  2 L a
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` v 0
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` v 0
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`  H
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`t  1 .5 T r
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`t  1 .5 T r
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`v 0
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`t  2 T r
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` H
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`T c  2 L a
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`
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`(1.1)
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`where a is the pressure wave-speed, V0 the initial flow velocity, g the acceleration due to
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`gravity, L the pipe length, and Tc the valve closure time. The time period,
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`, is the
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`time taken by the pressure wave to propagate down the pipe length, get reflected and
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`travel back. It is essential to consider the peak pressures due to water hammer in the
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`design of any pipeline system, which makes water hammer a well-studied topic in civil
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`engineering. Various researchers have simulated transient flow in pipeline systems with
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`different methods, a discussion of which follows in the next section.
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`Water hammer is a fast transient in the wellbore as compared to the conventional
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`pressure transient response of the reservoir. Water hammer, in the upstream petroleum
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`industry, has been a largely under-studied phenomenon. However, it has been a known
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`issue following emergency shut-ins of water injectors. Due to safety concerns, the
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`number of emergency shut downs of offshore injection wells can be high, with more than
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`80 emergency shut downs per year in some cases (McCarty and Norman, 2006). In recent
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`years, with the increasing number of offshore water injectors, it has been observed that
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`injection wells that undergo repeated shut-ins show reduced injectivity, higher sand
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`production and even failure of downhole completion (Vaziri et al., 2007). This
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`observation has been widely attributed to the cyclic pressure waves induced by water
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`hammer (Santarelli et al., 2000; Hayatdavoudi, 2006; McCarty and Norman, 2006; Vaziri
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`et al., 2007; Wang et al., 2008). It is believed that in weak sands, pressure fluctuations as
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`low as tens of psi, at the sand face, might be sufficient to cause sand failure (Santarelli et
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`al., 2000). Modeling work by Vaziri et al. (2007) have also shown that cyclic pressure
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`fluctuations cause more sanding than a monotonic increase in injection pressure.
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`
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`3
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` H 
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`aV0
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`g
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`2 L a
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`Figure 1.1: Conceptual schematic of water hammer in a reservoir-pipe-valve system
`(From KSB Know-how series on Water Hammer)
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`The magnitude of water hammer measured at the wellhead is often in the order of
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`hundreds of psi, however, there is almost no bottomhole water hammer pressure data in
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`injectors to confirm the magnitude at the sand face. It is, therefore, important to model
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`water hammer in injectors. The first objective of this study is to model water hammer in
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`injectors in order to estimate bottomhole water hammer pressures from measured surface
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`data.
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`It is also well known that sonic waves can be used to determine important
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`information about fracture and formation properties (Mathieu, 1984; Medlin, 1991). In
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`fact, it has been proved that fracture dimensions can be estimated from the propagation
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`and reflection of a single pressure pulse induced at the surface of a wellbore (Holzhausen
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`and Gooch, 1985; Paige et al., 1992). In principle therefore, it should also be possible to
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`extract similar information from the analysis of water hammer pressure waves. Moreover,
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`there is almost always a pressure gauge at the wellhead and water hammer pressure data
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`can be collected without any extra effort. Hence, any information that can be derived
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`from this data, independent of conventional testing methods, could be attractive and
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`useful to the oil industry. The second objective of this study is to develop a model to
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`estimate fracture connectivity and/or dimensions from water hammer pressure data.
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`1.1 LITERATURE REVIEW
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`1.1.1 Water hammer Modeling
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`Classical solutions of the basic unsteady flow equations were developed by
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`Allievi (1902, 1913) by analytical and graphical methods after neglecting the friction
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`terms. Bergeron (1935, 1936) also developed graphical solutions that were used
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`popularly before the advent of computers. Friction could be included by complex
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`procedures and for practical reasons the analysis was limited to single pipelines. Streeter
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`and Wylie (1967) proposed and popularized the explicit method of characteristics (MOC)
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`to solve the water hammer equations. Shimada and Okushima (1984) solved the water
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`hammer equations by a series solution method and a Newton-Rhapson method. Chaudhry
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`and Hussaini (1985) used MacCormak, Lambda and Gabutti Finite Difference (FD)
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`schemes to numerically solve the water hammer equations. Izquierdo and Iglesias (2002,
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`2004) developed a computer program using method of characteristics to simulate
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`transients in simple and complex pipeline systems. Silva-Araya and Chaudhury (1997)
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`solved the hyperbolic part of the equations in one-dimensional form by MOC and the
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`parabolic part in quasi-two-dimensional using finite difference. Ghidaoui et al. (2002)
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`proposed a two-layer and five-layer eddy viscosity model for water hammer where a
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`dimensionless parameter (the ratio of the time scale of the radial diffusion of shear to the
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`time scale of wave propagation) was used to estimate the accuracy of the assumption of
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`flow axisymmetry. Zhao and Ghidaoui (2003) have solved a model for quasi-two-
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`dimensional turbulent water hammer flow. Zhao and Ghidaoui (2004) have also
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`developed first and second-order Godunov-type explicit finite volume (FV) schemes for
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`water hammer problems. They have compared their schemes with MOC considering
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`space-line interpolation for three test cases with and without friction. They found that the
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`first-order FV schemes have the same accuracy as MOC with space-line interpolation but
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`for a given level of accuracy, the second-order scheme requires much less memory and
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`execution time than the first-order Godunov-type scheme. Wood (2005a, 2005b)
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`proposed the Wave Characteristic Method (WCM) and demonstrated that though, both
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`WCM and MOC, have the same level of accuracy, the WCM is more computationally
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`efficient for complex pipe systems. Greyvenstein (2006) proposed an implicit FD method
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`based on the simultaneous pressure correction approach. Afshar and Rohani (2008)
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`proposed a water hammer simulation using an implicit MOC scheme. It is evident from a
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`study of the previous work done that there are various numerical models such as explicit
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`and implicit Method of Characteristics, explicit and implicit finite difference, finite
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`volume and finite element to solve hydraulic transient problems. Among these methods,
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`the explicit MOC is the most popular for water hammer simulations for being simple to
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`code, accurate and efficient.
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`The general way of calculating friction losses in transient flows were using
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`formulae developed for steady-state conditions, for example the use of Darcy-Weisbach
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`equation for friction based on the mean flow velocity assumes that the shear stress at the
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`wall is the same for steady-state and transient flow conditions. The MOC solutions were
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`improved by incorporating unsteady or transient friction models instead of constant or
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`steady state friction used in the early models. Zielke (1968) proposed a convolution based
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`frequency dependepent model of unsteady friction for laminar flows that was very
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`computationally intensive. Trikha (1975) improved the computation speed of Zielke’s
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`model by using approximate expressions for Zielke’s weighting functions. Vardy and
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`Brown (2004) evaluated wall shear stress in unsteady pipe flows building on the previous
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`work by Trikha, but their solutions were faster and valid for both laminar and turbulent
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`flows. Vardy and Hwang (1991) adopted a five-region turbulence model and a different
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`expression in each region to compute the eddy viscosity distribution. Silva-Araya (1993)
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`incorporated an energy dissipation factor to compute laminar and turbulent unsteady
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`friction losses. Brunone et al. (1991) proposed a model where the total friction was the
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`sum of a qusi-steady friction and an unsteady friction that depended on the instantaneous
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`local and convective acceleration. Bergant et al. (2001) incorporated the two unsteady
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`friction models by Zielke (1968) and Brunone et al. (1991) into MOC and compared the
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`results against experiments. They found the Brunone model to be computationally
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`effective. Saikia and Sarma (2006) proposed a numerical model using MOC and unsteady
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`friction calculated at every time step using Barr’s (1980) explicit friction factor
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`correlation.
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`Water hammer in injectors have been modeled by Moos et al. (2006) and Wang et
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`al. (2008) as a Stoneley wave of amplitude given by Joukowsky’s formula (Eq. 1.1) that
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`propagates down the wellbore. Moos et al. (2006) have also demonstrated that using the
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`known physics of Stoneley wave propagation and attenuation in rocks, the formation
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`permeability and porosity can be estimated from water hammer data.
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`1.1.2 Fracture Impedance
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`Khalevin (1960), Walker (1962) and Morris et al. (1964) have used acoustic
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`waves to detect wellbore fractures. Mathieu (1984) derived analytically that Stoneley
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`waves could be used to detect hydraulic fractures by realizing that the presence of a
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`fracture changed the acoustic impedance of the wellbore. Mathieu derived the reflection
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`and the transmission coefficients for waves in fractured wellbore and introduced the term
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`“fracture impedance”. Hornaby et al. (1989) and Tang and Cheng (1989) subsequently
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`extended Mathieu’s work to vertical and horizontal fractures. Medlin (1991) introduced
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`tube waves (very low frequency Stoneley waves) to detect high permeability fractured
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`zones and the connectivity of such zones with the cased hole. Holzhausen et al. (1985,
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`1986) proposed that the altered acoustic impedance due to a fracture in the wellbore
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`could also be demonstrated and analyzed by the characteristics of pressure oscillations at
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`the wellhead. A single artificially induced pressure pulse at the surface would propagate
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`down the wellbore, get reflected and be transmitted back to the surface. This Hydraulic
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`Impedance Testing (HIT) used a lumped resistance-capacitance in series to model the
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`fracture and estimate the fracture impedance from the reflected pulse by trial and error.
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`The amplitude of the reflected pulse, which is determined by the impedance contrast
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`between the wellbore and the fracture, could be used to compute the width and height of
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`the fracture. Holzhausen’s model was experimentally validated by Paige et al. (1992) and
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`some field scale tests were also carried out (Paige et al., 1993; Holzhausen and Egan,
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`1986). Paige et al. (1992) showed that the pressure wave would reach the tip of the
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`fracture and proposed that the length of the fracture could be estimated, by measuring the
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`time lapse between the reflections of the wave from the entrance to the fracture and the
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`tip of the fracture. Ashour (1994) generalized Holzhausen’s HIT method for vertical and
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`horizontal hydraulic fractures and showed that sending a wave that is close to the
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`resonance frequency of the fracture can make a more accurate assessment of fracture
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`dimensions. Holzhausen’s model assumed no energy losses in the wellbore, which meant
`
`that the attenuation of the pressure wave due to friction in the wellbore was not taken into
`
`consideration. Patzek and De (2000) overcame this issue by proposing a lossy
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`transmission line model to describe the wellbore and fracture geometry and capture the
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`wellbore and fracture dynamics. In their model, flow through both the wellbore and the
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`fracture was treated analogous to the flow of electricity through transmission lines and
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`resistance, capacitance and inductance were distributed over the length of the line.
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`However, it has been the general opinion that it is difficult to collect the required
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`information from the measured pressure signal and HIT has not been used very popularly
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`in the industry.
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`9
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`EX_1060_020
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`Chapter 2: Model Formulation
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`2.1 WATER HAMMER MODELING EQUATIONS
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`The basic differential equations for transient flow in closed conduits are the one-
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`dimensional conservation equations of mass (continuity equation) and momentum
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`(equation of motion). The generalized forms of these equations were derived by using the
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`Reynolds transport theorem and then simplified using assumptions that are valid for
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`water hammer analysis (Chaudhury, 1987). Wylie and Streeter (1993) have also provided
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`a detailed derivation and discussion of these governing equations. The following sections
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`present a brief description of these equations from their work.
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`2.1.1 Continuity Equation
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`The general form of the continuity equation is
`
`
`
`
`
`(2.1)
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`where, A = area of cross-section of conduit, ρ = density of the fluid, V = mean flow
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`velocity, t = time, x = coordinate axis along the axis of the conduit. The first term in Eq.
`
`(2.1) accounts for the compressibility of the fluid and the second term represents the rate
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`of deformation of the conduit wall.
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`
`Assuming an elastic conduit filled with a slightly compressible fluid, Eq. (2.1)
`
`simplifies to
`
`
`
`
`
`(2.2)
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`where, p = pressure intensity, V = mean flow velocity, a = wave speed or the velocity of
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`the water hammer waves. For low-Mach-number unsteady flows, the transport term
`
`
`
`10
`
` 0
`
`
`
`V
`
`x
`
`dA
`
`dt
`
`1 A
`
`
`
`d
`
`dt
`
`1 
`
`p
`
`t
`
` V
`
`p
`
`x
`
` a 2 V
`x
`
` 0
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`EX_1060_021
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` is small compared to the other terms and may be dropped to yield the simplified
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`continuity equation
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`
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`2.1.2 Equation of Motion
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`
`
`(2.3)
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`The general form of the momentum equation is
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`
`
`
`
`(2.4)
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`where, f = Darcy-Weisbach friction factor, θ = angle of inclination of the pipe and, D =
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`diameter of the pipe.
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`Once again, the convective transport term,
`
`is neglected for low-Mach-
`
`number unsteady flows, reducing Eq. (2.4) to
`
`
`
`
`
`(2.5)
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`It is often convenient to analyze pipeline flows by defining pressure, p, in terms of
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`the piezometric head, H and use the discharge, Q, instead of the flow velocity, V.
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`
`
`
`
`(2.6, 2.7)
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`where, p = pressure, g = acceleration due to gravity, ρ = density of the fluid, z = elevation
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`of the pipe above a specified datum, V = mean flow velocity, and, A = area of cross-
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`section of the pipe.
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`Eqs. (2.3) and (2.5) expressed in terms of H and Q, become:
`
`
`
`11
`
`V p x
`
`p
`
`t
`
` a 2 V
`x
`
` 0
`
`fV V
`
`2 D
`
` 0
`
` g sin 
`
`p
`
`x
`
`1 
`
`
`
`V
`
`x
`
` V
`
`V
`
`t
`
`V V x
`
`fV V
`
`2 D
`
` 0
`
` g sin 
`
`p
`
`x
`
`1 
`
`
`
`V
`
`t
`
` z
`
`p 
`
`g
`
`H 
`
`Q  V A
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`(2.8, 2.9)
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`2.1.3 Velocity of Water Hammer Waves
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`The following general expression for the wave propagation velocity a, was
`
`presented by Halliwell (1963)
`
`
`
`
`
`(2.10)
`
`where, ψ = nondimensional parameter that depends on the elastic properties of the
`
`conduit, E = Young’s modulus of elasticity of the conduit walls, K = bulk modulus, and ρ
`
`= density of the fluid, respectively.
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`Expressions of ψ under various conditions (rigid conduit, thick-walled elastic
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`conduits, thin-walled elastic conduits, tunnels through solid rock, reinforced concrete
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`pipes, etc.) are available in the literature (Chaudhry, 1987; Wylie and Streeter, 1993). For
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`our analysis, we use the expression of ψ, valid for thin-walled elastic conduits anchored
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`against longitudinal movement throughout its length, given as
`
`
`
`
`
`(2.11)
`
`where, D = conduit diameter, e = wall thickness, ν = Poisson’s ratio of pipe material.
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`2.2 METHOD OF CHARACTERISTICS
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`The water hammer modeling equations are a pair of quasi-linear, hyperbolic,
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`partial differential equations and a closed-form solution of these equations is not
`
`
`
`12
`
`H
`
`t
`
`
`
`a 2
`
`Q
`
`g A
`
`x
`
` 0
`
`Q
`
`t
`
` g A
`
`H
`
`x
`
`
`
`fQ Q
`
`2 D A
`
` 0
`
`a 
`
`K
`
`
`
` 1  K E
`
`
`
`
`
`
`1  2
`
`
`
`D e
`
` 
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`available. However, there are several methods to numerically integrate these equations,
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`such as, method of characteristics, explicit and implicit finite-difference methods, finite-
`
`element methods, etc. Amongst these, the method of characteristics has been the most
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`popular due to its several advantages over other methods, particularly in water hammer
`
`type problems. These advantages include an explicit form of solution such that different
`
`elements can be solved independently and complex pipe networks can be handled with
`
`ease, an established stability criterion, an easy to program and computationally efficient
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`procedure and most importantly, accurate solutions. The main disadvantage of this
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`method is the requirement to adhere to the time step-distance interval relationship.
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`The momentum and continuity equations, in terms of two dependent variables,
`
`discharge and piezometric head, and two independent variables, distance along the pipe
`
`and time, are transformed into four ordinary differential equations by the method of
`
`characteristics. For further discussion let us rewrite the momentum and continuity
`
`equations (Eqs. 2.8 and 2.9) as
`
`
`
`
`
`
`
`(2.12, 2.13)
`
`A linear combination of these equations using an unknown multiplier λ yields
`
`
`
`(2.14)
`
`Please note that, using any two real, distinct values of λ, Eq. (2.14) will again
`
`yield two equations that are equivalent to Eqs. (2.12) and (2.13). Also, if
`
`
`
`and
`
`, then the total derivative can be written as
`
`
`
`13
`
`L1 
`
`Q
`
`t
`
` g A
`
`H
`
`x
`
`
`
`fQ Q
`
`2 D A
`
` 0
`
`L 2  a 2 Q
`
`x
`
` g A
`
`H
`
`t
`
` 0
`
` 0
`
`fQ Q
`
`2 D A
`
`
`
` 
`
`H
`
`t
`
`1 
`
`
`
`H
`
`x
`
` 
`
` g A
`
` 
`
` a 2 Q
`x
`
`Q
`
`t
`
` 
`
`L  L1  L 2 
`
`H  H ( x, t )
`
`Q  Q ( x, t )
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`IWS EXHIBIT 1060
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`EX_1060_024
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`(2.15, 2.16)
`
`It can be seen by from Eqs. (2.14), (2.15) and (2.16), that if λ is defined as
`
`
`
`(2.17)
`
`Then, by substituting the two particular values of λ, Eq. (2.14) can be written as two pairs
`
`of equations and identified as
`
`and
`
` equations.
`
`
`
`
`
`(2.18, 2.19)
`
`Thus, by imposing a relationship between the two independent variables, the
`
`original partial differential equations (Eqs. 2.8 and 2.9) were converted to two total
`
`differential equations. These ordinary differential equations, however, are not valid
`
`everywhere in the x-t plane like the Eqs. (2.8) and (2.9) were. Instead, Eq. (2.18) and Eq.
`
`(2.19) is only valid alo