End-Semester Test 2011

7/31/2019 End-Semester Test 2011

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End-Semester Examination

Numerical Reservoir Simulation - May 2011

Time: 2 hours + 10 minute reading time prior to start.

Open book test: all reference materials allowed including computers. Use of mobilephones, email and Skype not allowed.

The paper is divided into six Sections on the basis of the number of marks awarded foreach Exercise. Exercises may be completed in any order from all Sections.

Where the Exercise asks you to define the terms used, you must give a definition of eachterm used in an equation. For example, Darcys Law can be written as

q =kA(1 2)

L

An acceptable answer could define the terms as

q =flow rate.

k =permeability,

A =area,

=potential,

=viscosity,

L =distance.

If the terms are not defined, then no marks will be awarded!

Write your answers to the Exercises on the test paperbrief answers are best!

1


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1 SECTION - 50 MARKS TOTAL 2

1 Section - 50 marks total

Exercise 1.1. In a 3D cartesian grid how many near neighbour connections does a typicalgrid block have? (1 mark)

Exercise 1.2. In a 2D grid using hexagonal cells, how many near neighbour connectionsdoes a typical grid block have? (1 mark)

Exercise 1.3. Reservoir simulation solves equations that arise from conservation of mass.For compositional simulation, what does conservation of mass refer to? (1 mark)

Exercise 1.4. What is the molecular formula for propane? (1 mark)

Exercise 1.5. What is the molecular formula for carbon dioxide? (1 mark)

Exercise 1.6. Is hydrogen sulphide, H2S, a hydrocarbon or non-hydrocarbon component?(1 mark)


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Exercise 1.7. In compositional simulation, how many components make up the oil phase?(1 mark)

Exercise 1.8. In compositional simulation, how many components make up the waterphase? (1 mark)

Exercise 1.9. Write an equation for fractional flow for oil-water (two-phase) flow anddefine the terms used. (1 marks)

Exercise 1.10. Define pore volume. (1 mark)

Exercise 1.11. In reservoir simulation, we normally take into account the change in pore-volume with pressure, but not the change in total (or gross) rock volume with pressure.Why is this? (1 mark)


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Exercise 1.12. If Sg > 0 and So > 0, the 5 typical unknowns in black-oil simulation,po, Sw, Sg, Rs and rs reduce to 3 unknowns. What are they? (1 mark)

Exercise 1.13. Does the fluid within a porous medium move faster, slower or at the samevelocity as the Darcy velocity? (1 mark)

Exercise 1.14. Which geoscience discipline usually generates the top structure map of areservoir? (1 mark)

Exercise 1.15. In converting a seismic time-map to depth, what property of the rockbetween the surface and mapped formation is used? (1 mark)

Exercise 1.16. Write an equation for the mass of component i in a multiphase systemwhere Sj =volume fraction of phase j; j =density of phase j; V =total volume; wij =massfraction of component i in phase j. (1 mark)


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Exercise 1.17. If c =pore compressibility and cj =phase compressibility, write an equa-tion for total compressibility ct in terms of these quantities and phase saturations Sj.(1

mark)

Exercise 1.18. Assuming water-wet conditions, how does water-oil capillary pressurevary with water saturationdoes it increase or decrease with increasing water saturation?(1 mark)

Exercise 1.19. Assuming that oil is the wetting-phase, how does oil-gas capillary pressurevary with oil saturationdoes it increase or decrease with increasing oil saturation? (1mark)

Exercise 1.20. Is the viscosity of a fluid flowing through a porous medium dependent insome way on the rock type of the porous medium? (1 mark)

Exercise 1.21. What is the shape of a fractional flow curve if a shock-front does notdevelop during two-phase flow? Sketch an example. (1 mark)


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Exercise 1.22. What is meant by mass balance? (1 mark)

Exercise 1.23. Denoting the bulk volume of a cell by VB, the pore volume Vp, porosityby , oil phase volume by Vo and oil saturation by So, what is the relationship betweenthese? (1 mark)

Exercise 1.24. In a 3-dimensional simulation cartesian grid model, how many neighboursdoes a corner-cell have? (1 mark)

Exercise 1.25. For a 10 15 8 fully implicit simulation model, how many unknownsare there to be solved at each time-step? (1 mark)

Exercise 1.26. For a 20 cell 1-D simulation model using an IMPES solution method,what is the approximate ratio between the number of values stored in the full pressurematrix and the number of values stored in sparse matrix format? (1 mark)


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Exercise 1.27. Referring to the following figure,

at capillary equilibrium write down the relationship betweenp+, p, , , andR. (1 mark)

Exercise 1.28. What is the surface tension of n-Octane at 20C with respect to air? (1mark)

Exercise 1.29. Define the interfacial tension between oil and water in terms of theirsurface tensions. (1 mark)


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Exercise 1.30. What is Antonoffs relationship? (1 mark)

Exercise 1.31. What is wettability? (1 mark)

Exercise 1.32. What is an oil ganglion? (1 mark)

Exercise 1.33. If a gas has molecular weight of 18.7, what is its gas gravity? (1 mark)

Exercise 1.34. If an oil has a specific gravity of 0.82, what is its API gravity? (1 mark)

Exercise 1.35. Does the presence of capillary pressure increase or decrease the fractionalflow of water? (1 mark)


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Exercise 1.36. What is meant by a flash calculation at pressure p and temperature T?(1 mark)

Exercise 1.37. What is the type of grid in the following figure called? (1 mark)

Exercise 1.38. What is the type of grid in the following figure called? (1 mark)


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Exercise 1.39. What is a typical mole fraction of methane for a black oil? (1 mark)

Exercise 1.40. On the following P-T diagram, what is the name of region between thebubble-point line and the dew-point line called? (1 mark)

Exercise 1.41. Define the dew point line? (1 mark)

Exercise 1.42. What is the range of formation volume factors for typical black oils? (1mark)


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Exercise 1.43. When approximating a derivative with a finite-difference approximation,we write the equation p

x= p(x+x)

p(x)+ (x). What is the term (x) called? (1 mark)

Exercise 1.44. What does round-off error mean? (1 mark)

Exercise 1.45. What does numerical dispersion mean? (1 mark)

Exercise 1.46. If the size of the grid cells in a simulation model increases, does numericaldispersion increase or decrease? (1 mark)

Exercise 1.47. What is a curvilinear grid? (1 mark)


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Exercise 1.48. What does the term LIFT TABLE refer to in reservoir simulation? (1mark)

Exercise 1.49. What are non-neighbour or special connections? (1 mark)

Exercise 1.50. What is meant by the expression equivalent well block radius? (1 mark)


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Exercise 2.1. Sketch the pore doublet model of a porous medium showing connate water.(2 marks)

Exercise 2.2. If a grid cell has dimensions x by y, what is an expression for theequivalent well block radius re in terms of x and y? (2 marks)

Exercise 2.3. What does Cost-Benefit analysis mean? (2 marks)

Exercise 2.4. What are the two main items to match during history matching of a 2-phase oi-water reservoir? (2 marks)


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Exercise 2.5. There are two types of numerical method commonly used for solving thereservoir simulation equations which take into account the stability of the solution with

respect to time-step size. What are these methods called? (2 marks)

Exercise 2.6. Generally there are three phases in a reservoir: the water, oil and gasphases. The water phase is commonly referred to as the aqueous phase. What are thecommon alternative names for the oil and gas phases? (2 marks)

Exercise 2.7. What is the general molecular formula for the napthene series? (2 marks)

Exercise 2.8. What does coning mean? (2 marks)

Exercise 2.9. Describe two geological features where we may need to use non-neighbouror special connections in a simulation model? (2 marks)


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Exercise 2.10. If we write the following equations for the mass of oil no and the massof gas ng in a simulation cell at reservoir conditions

no =SoVp

Bo+

rsSgVpBg

ng =RsSoVp

Bo+

SgVpBg

,

write equations for the volume of oil Vo = SoVp and the volume of gas Vg = SgVp. (2marks)

Exercise 2.11. A 1-dimensional single phase simulation model has the following values ofthe parameters ct = 7 10

6 psia1, x = 120 ft, k = 250 mD, = 0.4 cp and = 0.18.

What is the maximum stable time-step size for a rate-constrained explicit solution to thefinite difference equations? (2 marks)

Exercise 2.12. One calculation of the undersaturated formation volume factorBo is givenas a function of both p and ps by

Bo = Bosexp(cos(pps))

An alternative is given by

Bo = Bos ppscosps


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Given that saturation pressure is 700 psia, reservoir pressure is 2100 psia, oil formationvolume factor at saturation pressure is 1.25, and oil compressibility at saturation pressure

is 12 106

psi1

, calculate the oil formation volume factor at reservoir conditions usingboth methods. Which is more likely to be correct? (2 marks)

Exercise 2.13. How many non-zero cells are there in the matrix used to solve an implicit3 4 2-dimensional single-phase model (with rate constraints)? (2 marks)

Exercise 2.14. If capillary pressure is not negligible, give an equation for the fractionalflow of water and oil in terms of oil and water relative permeability and oil and waterviscosity, and capillary pressure. (2 marks)

Exercise 2.15. What hydrocarbon components make up kerosene? (2 marks)


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Exercise 2.16. Show using diagrams, two isomers of pentane. (2 marks)

Exercise 2.17. Define the total formation volume factor for two-phase gas-oil systems.(2 marks)

Exercise 2.18. The Glaso correlation for oil formation volume factor is given by

log(Bos 1) = 6.58511 + 2.91329(logBo) 0.27683(logB

o)

2

logBo = Rs

go

0.526+ 0.968T

given oil gravity = 0.83, gas gravity = 0.95, pressure is 1310 psia, solution gas content= 450 scf/bbl, and temperature = 182 F, what is the oil formation volume factor? If thesolution gas content increases, does the oil formation volume factor increase or decrease?(2 marks)


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Exercise 2.19. What is meant by a recombination oil PVT sample? (2 marks)

Exercise 2.20. What does a constant composition expansion experiment measure? (2marks)

Exercise 2.21. What does a differential liberation experiment measure? (2 marks)

Exercise 2.22. In a depletion drive reservoir, why are oil PVT properties derived fromdifferential liberation used in reservoir simulation rather than constant composition ex-pansion? (2 marks)


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Exercise 2.23. Why do we need to correct laboratory measured PVT properties for fieldconditions? What do we mean by this? (2 marks)

Exercise 2.24.

What is meant by shrinkage factor when refering to different separatortests? (2 marks)

Exercise 2.25. In the following table

which separator pressure and temperature results in the lowest oil shrinkage, and whichpressure and temperature results in the highest shrinkage? (2 marks)


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Exercise 2.26. Write down the Redlich-Kwong equation of state and explain the terms?(2 marks)

Exercise 2.27.

If f

V

i is the fugacity of the i

th

component in the vapour phase, and f

L

i isthe fugacity of the ith component in the liquid phase, what is the relation between these ifthe phases are in chemical equilibirum. (2 marks)

Exercise 2.28. If a gas reservoir is in contact with an aquifer, the interface between thegas and the water is defined as the gas-water contact (GWC). Suppose that the aquiferis depleted by fluid withdrawals from another fieldthat is, the pressure in the aquifer

decreases, but there is no production from the gas reservoir itself? Does the GWC 1)remain at the same depth, 2) increase in depth (that is, deeper), or 3) decrease in depth(that is, shallower)? (2 marks)

Exercise 2.29. Oil compressibility can be calculated as

co = 1Bo

Bop

+ BgBo

Rsp

Explain in words why we need to include the term BgBo

Rsp

. (2 marks)


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Exercise 2.30. Consider the following diagram showing the distance between the centresof two simulation cells.

If the width of each cell is y and the vertical thickness is z, and the permeabilities ofthe cells are ki and ki+1, respectively, write an equation for the transmissibiluty betweenthe two cells. (2 marks)


Exercise 3.1. We talked about the zone of consistency in the introductory lecture andin the lecture on history-matching. What are the three factors whose intersection creates

the zone of consistency? (3 marks)


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Exercise 3.2. Give three examples of different types of reservoir simulator. (3 marks)

Exercise 3.3. There are five basic mechanisms for recovering oil from reservoirs, two ofthese are fluid expansion and fluid displacement. Name the other three. (3 marks)

Exercise 3.4. Hydrocarbon components such as methane form part of the compositionof the oil and gas phases in the reservoir. Name three non-hydrocarbon components. (3marks)

Exercise 3.5. It is planned to sequester carbon dioxide CO2 in an aquifer. If we wish totake into account the formation of carbonates in the reservoir due to interation between

the CO2 and minerals in the rock, what type of simulation model should we use? (3 marks)


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Exercise 3.6. What assumption about the impact of temperature in the reservoir is usu-ally made in 1) black-oil simulation, 2) compositional simulation, and 3) thermal simula-

tion? (3 marks)

Exercise 3.7. A tank model assumes that the pressure and temperature is everywhere

constant in the reservoir. Explain why this assumption is often suitable for gas reservoirs,but is unlikely to be suitable for oil reservoirs. (3 marks)

Exercise 3.8. Let z be the depth difference between two grid cells each filled with oilof density o at reservoir conditions. If the initial oil pressure in each cell is p1 and p2,respectively, what is the relationship between p1, p2, and z to ensure equilibrium? (3

marks)


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Exercise 3.9. If we have defined the reservoir geometry and geology, rock properties andfluid properties, give at least three other items that we need to define before we can proceed

with running a simulation model. (3 marks)

Exercise 3.10. In an oil-water reservoir, decreasing the pressure results in compaction

of the reservoir rock, which leads to reduced pore-throat size and reduced permeability.Explain why the usual formulation of a black-oil simulation model is not suitable to forthis case. What do we need to include in the formulation to make the model suitable forthis reservoir? (3 marks)

Exercise 3.11. If Sg > 0 and So > 0 then we have saturated oil and gas. The un-knowns in the simulation equation are typically po, Sw and Sg (oil pressure, water satu-ration and gas saturation, respectively). If So = 0 then there is no dissolved gas in the oilphase: what are the unknowns in the simulation equations in this case? (3 marks)


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Exercise 3.12. Briefly explain the terms surface tension, interfacial tension and capillarypressure. (3 marks)

Exercise 3.13. When could a reservoir which initially has only a gas phase and waterphase present, be modelled with a 2-phase reservoir simulator? (3 marks)

Exercise 3.14.

Consider a simple drinking straw. What does capillary pressure dependon? (3 marks)

Exercise 3.15. The diffusivity equation can be written in the form pt

= a 2p

x2 + q +J(ppw). What do the terms q and J(ppw) represent? (3 marks)

Exercise 3.16. What is the central difference approximation to the first derivative ofa function p with respect to a variable x ? (3 marks).


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Exercise 3.17. List three sources of error that arise in building a reservoir simulationmodel which are related to the formulation of the finite difference equations. (3 marks)

Exercise 3.18.

List three sources of error that arise from the size and shape of the gridcells used in numerical simulation? (3 marks)

Exercise 3.19. Oil and water relative permeability are calculated using the Corey formu-lation. That is, letting

Sw =Sw

1 Swc Sor(3.1)

then, if 0 Sw

1,

kro = kroe(1 Sw)

nco

krw = krwe(Sw)

ncw(3.2)

if Sw 0,

kro = kroe

krw = 0(3.3)


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if 1 Sor Sw 1,

kro = 0

krw = 1 (1 krwe)(1 Sw)

Sor

(3.4)

else, if 1 Sw,

kro = 0

krw = 1(3.5)

where

Swc = connate water saturation

Sor = residual oil saturation

kroe = oil relative permeability end-point

krwe = water relative permeability end-point

nco = Corey exponent for oil

ncw = Corey exponent for water.

If Swc = 0.25, Sor = 0.16, kroe = 0.7, krwe = 0.35, nco = 2.2 and ncw = 3, calculatethe oil and water relative permeabilities for 1) Sw= 0.12, 2) Sw= 0.35, 3) Sw= 0.95. (3marks)


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Exercise 3.20. Counter current flow occurs where oil and water flow in opposite direc-tions. Explain whyin reservoir simulationthe water relative permeability and the oil

relative permeability are typically calculated at different water saturations for counter-current flow between two grid cells. (3 marks)

Hint: consider which is the upstream cell for calculating the flow of each phase.


Exercise 4.1. Isothermal compressibility of water has been defined in terms of waterdensity. Derive the equation for isothermal compressibility in terms of the water formationvolume factor, Bw. (4 marks)

Exercise 4.2. List four examples where the matrix solved in reservoir simulation doesnot have the same number of non-diagonal entries as would be assumed in the normalformulation of the equationsthese have been referred to as symmetry breakers. (4marks)


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Exercise 4.3. By drawing a picture, explain why including faults in reservoir simulationusually results in the what are called non-neighbour or special connections. (4 marks)

Exercise 4.4. List four observations which may mean that we should include an aquiferin the simulation model. (4 marks)

Exercise 4.5. In the following figure, the symbol F shows the fill-in which occurs if thewe solve the matrix equation using a direct method.


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In the next figure, the equations have been re-ordered (D4 order) with the fill-in againshown by the symbol F.

Why would using D4 ordering reduce the time it takes to solve the equations on a com-

puter? (4 marks)


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Exercise 5.1. A differential liberation experiment and separator test on a sample of reser-voir oil are undertaken. The saturation pressure of the oil sample is 2550 psia. At thispressure the oil volume in the PVT cell relative to the residual oil volume at the end ofthe differential liberation experiment is 1.62 (bbl/resid bbl) and the GOR is 833 (scf/residbbl). At 2350 psia the ratio of the oil volume to residual volume is 1.57 (bbl/resid bbl)andthe GOR is 776 (scf/resid bbl). The results of the separator test are given in the followingtable:

Separator Stock Tank Shrinkage GOR

psia oF psia oF stb/rb scf /stb

350 80 14.7 60 0.6610 583

250 80 14.7 60 0.6513 573

135 80 14.7 60 0.6642 586

90 80 14.7 60 0.6766 599

What are Bo and Rs at the saturation pressure of the oil sample, and at 2350 psia,corrected for best separator conditions? (5 marks)

Exercise 5.2. Name five iterative solution methods that could be used to solve the equa-tions in a reservoir simulator. (5 marks)


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Exercise 5.3. Grid block orientaton can lead to physically unreasonable saturation frontsdeveloping in the simulation model as shown in the following figure.

Explain why this occurs and when could it be significant in terms of oil and water proper-ties. (5 marks)


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Exercise 5.4. Numerical dispersion can be seen to arise from the fact that the first orderapproximation to the saturation derivative

S

x=

S(x) S(xx)

x+ (x)

has order x, whereas the approximation

S

x=

S(x) S(xx)

x

x

2

2S

x2+

(x)2

has order x2. Explain why this leads to numerical dispersion in the model when thegrid cell size x increases. (5 marks)


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Exercise 6.1. Assuming the following matrix formulation of the explicit solution to thefinite difference equations for rate constrained wells pk+1 = Bpk + t q, show why super-position is true for two wells of rates q1 and q2, respectively.

Exercise 6.2. Why doesnt superposition work for pressure constrained wells? That is,where we have pk+1 = Bpk + J

pk pbhp

, for two wells with constant bottom-hole

pressures p1 and p2, respectively. (6 marks)


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Exercise 6.3. For a 2-dimensional single-phase model with well rate constraints, writedown the implicit finite difference equation for 1) a corner cell; 2) an edge cell; 3) a centre

cell. Hint. Make sure that the coefficient of the diagonal is correct. (6 marks)

Exercise 6.4. By refering to the following diagram

derive the equation for the effective block radius re as a function of x. (6 marks)


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Exercise 6.5. Briefly explain what each of the following inputs to reservoir simulationmeans:

1. Reservoir geometry

2. Simulation grid

3. Aquifer properties

4. Rock properties

5. Fluid properties

6. Rock-fluid interactions7. Fluid-fluid interactions

8. Well data

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End-Semester Test 2011