SPE-173465-STU

Analysis of Reservoir Pressure Transients: A Wave Physics Approach

Ohazuruike Lotanna Vitus, a 500L student of Department of Petroleum Engineering Rivers State University ofScience and Technology, Port Harcourt

Copyright 2014, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE international Student Paper Contest at the SPE Annual Technical Conference and Exhibition held in Amsterdam,The Netherlands, 2729 October 2014.

This paper was selected for presentation by merit of placement in a regional student paper contest held in the program year preceding the International Student PaperContest. Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). Thematerial, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members.Electronic reproduction, distribution,or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted toan abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

This research work presents an investigation on a wave physics approach to pressure transient analysis.It is hinged on the view that a pressure transient phenomenon has an associated wave behaviour. AReservoir pressure transient partial differential equation is developed by importing the pressure transientvelocity into the general wave partial differential equation. The equation is then solved using theBoltzmann transformation. The resulting equation presents a mathematical definition of a reservoirpressure transient from a wave physics viewpoint. Spectral and Time Series plots of the model and fielddata predicted pressure transients are presented. Observation shows a similar trend in behaviour for bothplots and discrepancies possibly due to reservoir heterogeneities, noise in data and the likelihood of acounterpressure effect from the restrictions.

IntroductionThe idea of matter waves was presented by Louis de Broglie in 1922 1. As he remarked, a movingparticle, whatever its nature, has wave properties associated with it 2. A direct consequence of this is thatthe motion of matter, oil inclusive, is inseparable from wave motion. This has been the idea behind whatis referred to as the Wave Mechanics Model 2, 3.

One of the most important aspects of fluids is the wide variety of waves which can be generated andsustained in them 3. If a pebble were dropped in the middle of a river, one observes the generation anddissemination of waves from the source outwards, towards the water boundary. Described like this, apressure transient (or pulse) is a wave phenomenon [disturbance travel] that accompanies a rapid changeof the velocity of the fluid in the medium. The travel of these transients in a reservoir invariably involvesthe creation and subsequent transmission of pressure sink points through the reservoir, a phenomenon towhich the continual change in the Bottom Hole Pressure (BHP) of a well is attributed. Consequently, mostPTA studies involve the use of BHP data 4, 5.

Pressure transient responses of wells are often computed with numerical models by using fine griddingand very short time intervals 6. An alternative to this was proposed by Medeiros et al 6. The semi-analytical approach presented was reported to be more adequate for computing pressure transients for

more complex forms of heterogeneity including composite, layered and compartmentalized reservoirs, arenowned source of irregularities in Pressure Transient Analysis (PTA) 7.

Furthermore, the dependence of these pressure transient responses on the reservoir properties has alsomade it an indispensable tool for reservoir characterization. This invaluable contribution, amongst others,has made real-time pressure and production data monitoring and analysis inevitable 8. A complete reviewof its evolution over time, the different approaches proposed and employed thus far has been presentedby Gringarten 9.

Objective of studyThe objective of this study is to show mathematically that a reservoir pressure transient behavior is a wavephenomenon. The theory is also supported by observed Time series and spectral plots of the predictedpressure transients.

Limitation of studyThis research basically involves the treatment and analysis of pressure transients as waves, mathematicallyand via the use of spectral plots. Impacts of reservoir heterogeneities and the contributions from associateddiffusion phenomena are not considered.

Mathematical ModelingThe mathematical modeling proceeds in the following order:

1. Development of a reservoir pressure transient velocity equation;2. Development of a reservoir pressure transient wave equation by importing the pressure transientvelocity term into the radial form of the general wave equation;

3. Solving the developed reservoir pressure transient wave equation using the Boltzmannstransformation.

The wave equation in cylindrical coordinates, neglecting the vertical component is given by 10:

1.1

The term v in Equation 1.1 is referred to as the speed of the wave.it is important to note thatirrespective of the fact that the speed of a wave is related to the waves wavelength and frequency, it isset primarily by the properties of the medium 11.

The distance over which a pressure disturbance will travel during an elapsed time, t, is12;

1.2

The Boltzmann transformation is given as 13;

1.3

Model developmentDifferentiating Equation 1.2 with respect to time,t, gives;

2.1

Thus, a pressure transients velocity is given as;

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2.2

Developing the pressure transient wave equation basically entails importing the pressure wave velocityas defined in Equation 2.2 into Equation 1.1 to give;

2.3

This Equation is then solved using the Boltzmann transformation as given in Equation 1.3 and thefollowing Initial and Boundary Conditions 14;

i. P Pi at t 0, for all rii. P Pi at r , for t 0iii.

The Boltzmann transformation is given as

1.3

So that,

2.4

And

2.5

Modifying Equation 1.1 using the chain rule of differentiation,

2.6

Substituting Equation 2.5 in Equation 2.6,

2.7

2.8

2.9

But,

Substituting into Equation 2.9 gives,

2.10

Substituting Equation 2.10 in Equation 2.9

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2.11

Factoring out 1/t from the RHS term in square brackets,

2.12

Putting ,

2.13

Putting in the LHS of Equation 2.13,

2.14

But ,

2.15

Substituting and expanding the term in parentheses in the LHS of Equation 2.15,

2.16

This reduces to

2.17

Dividing both sides by ,

2.18

Opening the parentheses,

2.19

Taking like terms,

2.20

Putting ,

2.21

Taking like terms and integrating,

2.22

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From Boundary Condition,

2.23

Putting Equation 2.23 in Equation 2.22,

2.24

Integrating,

2.25

But, from total derivative,

2.26

Substituting Equation 2.26 in Equation 2.25 and adjusting the limits accordingly,

2.27

For dimensional accuracy,

2.28

which gives,

2.29

which in field units would be;

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2.30

MethodologyCase Study: Well X

Data acquired for Well X in the Niger Delta region of Nigeria were BHP readings taken at atwo-minute interval from a downhole pressure gauge. The BHP data acquired was for an eight-monthperiod. Also acquired was the production data for that Eight month period and the reservoir parameters.

The data preparation basically involved generating hourly and daily averages of the BHP valuesobtained, computing the change in successive BHP values and the cumulative time in hours and days atwhich these changes were observed.

The production rate and reservoir properties data were used to compute the pressure transients aspredicted by the model equation.

The daily changes in BHP data and the pressure transient computations from Equation (2.30) were usedto prepare normalized spectral plots using the Time Series tool of MATLAB 15. These plots were preparedfor two model scenarios and the calculated pressure transients from the BHP data. The pressure transientsare calculated on the assumption that a change in BHP effected by a pressure transient of an equalmagnitude. The two model scenarios differ on the value assumed for the Initial Pressure Transient term,

Figure 1aTime Series plot for first scenario of Model

Figure 1bTime Series plot for second scenario of Model

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Pi. The first scenario assumes a zero value for Pi whereas the second takes the preceding PressureTransient value as Pi.

Results and DiscussionFigures 1a and 1b and 2a and 2b show the time series and normalized spectral plots for the two modelscenarios respectively, as discussed. Close observation shows that there is almost no difference in theplots. Consequently, subsequent analysis employs one of the approaches.

Figures 3a and 3b above shows the time series plots for the zero Pi -model [chosen scenario] and thechange in BHPs calculated from the field data. A close observation shows that the model tends toover-predict the magnitude of the pressure transients. However, similar arrival of pressure transients isobserved from the near-correspondence of pressure transient peak points in the plots.

Figures 4a and 4b show the normalized spectral plots for both the zero Pi model and the calculatedpressure transients. Observation reveals that the model is most accurate in the earlier period as can beobserved from the appearance of a similar range off peak points on both plots. However, it is observedthat the models peak range of pressure transients appear a while after that of the data. This lapse could

Figure 2aNormalized spectral plot for first scenario of Model

Figure 2bTime Series plot for second scenario of Model

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Figure 3aTime Series plot for first scenario of Model

Figure 3bTime Series plot for Field data

Figure 4aSpectral plot for first scenario of Model

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be attributed to a host of factors like reservoir heterogeneity, inaccurate drainage radius, amongst others.itis also observed that though the magnitudes be different, a similar drop is experienced towards the end.

SummaryThis work has presented a mathematical proof that contrary to the view of renowned academics 12, thewave PDE could be solved to obtain a model describing the travel of reservoir pressure transients. It alsorevealed that the logarithm solution, declared an approximation by the diffusivity theorem, is in fact anactual solution according to the wave theorem, a view supported by field observations.

Conclusion and RecommendationTime series and spectral plots of the developed model and field data pressure transients have beenpresented. The results show that though the model over-predicts the magnitude of the pressure transient,similar trends are observed. As such, further experimentation and approaches to the use of this theory isrecommended.

AcknowledgmentThe contributions of Engr. Cyrusba Dagogo-Jack and Engr K.K. Dune in the supervision of this work arehugely appreciated.

Nomenclatures

P Pressure, psiQ flow rate, bpd viscosity, cpB formation volume factor, rb/stbK permeabilityH reservoir thicknessT time, hrs porosity,C compressibility, psi-1

R radius, ft

Figure 4bSpectral plot for Field Data

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10. Riffe, D.M. (2013) The Wave Equation in Cylindrical Coordinates, Utah State University[online], available at http://www.physics.usu.edu/riffe/3750/Lecture%2020.pdf (Accessed 14July, 2013)

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12. Zimmerman, R.W. (2002) Fluid Flow in Porous Media, Imperial College London [online],Available at http://workspace.imperial.ac.uk/earthscienceandengineering/Public/external/Staff/RobertZimmerman/Fluid%20Flow%20in%Porous%20Media.pdf (Accessed 14 July, 2013)

13. Dake, L.P. (1998) Fundamentals of Reservoir Engineering, Amsterdam, Elsevier, pp. 12722414. Ahmed, T. (2010) Reservoir Engineering Handbook, 4th Edn, Massachusetts, Gulf Professional

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