exam 2011

English NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET INSTITUTT FOR PETROLEUMSTEKNOLOGI OG ANVENDT GEOFYSIKK Kontaktperson: Jon Kleppe, mobil 91897300 Institutt 73594925

SOLUTION

FINAL EXAM IN COURSE TPG4150 RESERVOIR RECOVERY TECHNIQUES

Monday December 12, 2011

Time: 0900-1300

Allowed material: -Type-approved calculator, with empty memory, according to NTNU´s list of approved calculators is allowed. -No printed or hand-written materials permitted. The grading will be ready on January 6, 2012

TPG4150 Reservoir Recovery Techniques Final exam 12.12.11

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Symbols used are defined in the enclosed table Question 1 (10 points) This question relates to the Gullfaks project work. a) Please outline (briefly) the process of investigation of the K1/K2 segment of the Gullfaks field

using Material Balance analysis. • By use of MBE based on the volumes and reservoir rock and fluid properties of the

K1/K2 segment, and the reported production and injection data, the theoretical pressure development may be compared with the real (reported) pressure development, and thus give insight into the communication between the segment and the surrounding reservoir.

b) What type of measured parameters, both initially and during production, are normally available for a real reservoir system • Reservoir rock and fluid properties, initial volumes and fault communication estimates,

production and injection rates, and wellhead pressures. c) Based on your experience of the project work, list the major uncertainties normally encountered

when addressing a real reservoir system. • Effective reservoir volumes (pore volume, oil, gas cap, water volumes) • fault communication • communication with surrounding reservoir • aquifer strength

Question 2 (15 points) - 2-2008 Write or derive an expression (equation or text) that defines each of the following terms (see list of symbols at the back): d) Formation volume factor e) Solution gas-oil ratio f) Fluid compressibility g) Pore compressibility h) Total reservoir compressibility i) Expansion volume (approximate) due to compressibility and pressure change j) Real gas law for hydrocarbon gas k) Reservoir oil density l) Reservoir gas density m) Reservoir water density n) Relationship between oil compressibility (undersaturated) and formation volume factor o) An expression for gas compressibility using the real gas law p) What do we mean with "microscopic" and "macroscopic" recovery factors? q) How can we improve the "microscopic" recovery of a reservoir? r) How can we improve the "macroscopic" recovery of a reservoir? solution---15x1 points---

a) Formation volume factor

�

B = (res.vol.)(st.vol.)


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b) Solution gas-oil ratio Rso =

(st.vol. gas)(st.vol. oil)

c) Fluid compressibility cf = −1Vf

(∂Vf

∂P)T

d) Pore compressibility cr =1φ( ∂φ∂P)T

e) Total reservoir compressibility cT = cr + cii=o,w,g∑ Si

f) Expansion volume (approximate) due to compressibility ΔV = V2 −V1 ≈ −V1c(P2 − P1) g) Real gas law for hydrocarbon gas PV = nZRT

h) Reservoir oil density ρoR =ρoS + ρgSRso

Bo

i) Reservoir gas density ρgR =ρgS

Bg

j) Reservoir water density ρwR =ρwSBw

k) Relationship between oil compressibility (undersaturated) and formation volume factor

�

Co = −1Vo

dVo

dP, and Vo = VoSBo

Thus,Co = −1Bo

dBo

dP

l) An expression for gas compressibility using the real gas law

Cg = − 1Vg

dVgdP, and PVg = nZRT

Thus, Cg =1P− 1ZdZdP

m) What do we mean with "microscopic" and "macroscopic" recovery factors? • microscopic is related to the end point residual saturation, as seen on relative

permeability curves • macroscopic is related to large-scale recovery factors mainly influenced by layering,

heterogeneity, well coverage, etc. n) How can we improve the "microscopic" recovery of a reservoir?

• By reducing interfacial tension between rock and fluids, eg. by surfactant additions to the injection water

o) How can we improve the "macroscopic" recovery of a reservoir? • By better volumetric sweep, through better well coverage, blocking of thief zones, etc.

Question 3 (13 points) For displacement of oil by water in a reservoir cross-section, answer following questions: Solution---13x1 points---


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a) What does the term "segregated flow" mean, and which factors determine this flow condition?

• Fluids separate according to density, and the flow is segregated flow if gravity gradients dominate the flow

ie.

�

gΔρ >>δPδx

b) What does the term "diffuse flow" mean, and which factors determine this flow condition? • Fluids do not separate according to density, and the flow is diffuse flow if dynamic

pressure gradients dominate the flow

ie.

�

δPδx

>> gΔρ (leads to uniform saturation distribution vertically)

c) What does the term "vertical equilibrium" mean in reservoir analysis, and when is it a realistic assumption? • Fluids segregate vertically immediately (in accordance with capillary pressure), and may

be realistic in high-permeability reservoirs with small dynamic gradients

ie. gΔρ>>

∂P∂x

(the “ultimate” segregated flow)

May be a reasonable assumption in high permeability reservoirs where dynamic gradiens are small and vertical segregation takes place quickly d) Sketch typical saturation profiles (in vertical direction) for “diffuse flow” conditions and

“segregated flow” conditions.

e) What does the term "piston displacement" mean in reservoir analysis, and when is it a realistic

assumption? • All movable oil is displaced immediately; require a very low mobility ratio

f) What is the Dykstra-Parsons method used for, and which assumptions are made for the method? • Displacement in layered systems without communication • Assumptions

o Constant pressure drop for all layers o piston displacement o capillary pressure negligible o No communication between layers

g) For a real reservoir, how realistic is the Dykstra-Parsons method? • piston displacement requires a very favourable mobility ratio, and gravity segregation

within layers may be significant. Also, communication between layers may be significant


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h) What is the Buckley-Leverett method used for, and which assumptions are made for the method? • Displacement calculations under diffuce flow conditions • Assumptions

o diffuse flow conditions o no capillary dispursion at front o incompressible fluids

i) For a real reservoir, how realistic is the Buckley-Leverett method? • For a high-permeability reservoir the condition of diffuse flow may not be realistic

j) What is the Dietz method used for, and which assumptions are made for the method? • stable displacement in inclined systems • Assumptions

o vertical equilibrium o piston displacement o negligible capillary pressure

k) For a real reservoir, how realistic is the Dietz method? • instant vertical segregation of fluids may not be realistic for medium to low-permebility

reservoirs l) What is the Vertical Equilibrium (VE) method used for, and which assumptions are made for

the method? • It is used for recovery calculations assuming instant segregation of fluids, ie. that the

vertical saturation distribution is known • Assumptions

o instant segregation of fluids

o gΔρ>>

∂P∂x

m) For a real reservoir, how realistic is the VE method? • It requires instant segregation of fluids, which may not be realistic in medium to low-

permeability reservoir Question 4 (12 points) a) Sketch a typical Dietz analysis displacement situation. Include interface, angles and flow

direction

α

β


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b) Sketch a typical Dykstra-Parson’s analysis displacement situation. Include layers, fluid fronts and flow direction

c) Sketch a typical Buckley-Leverett analysis displacement situation. Include a typical saturation

distribution profile.

S w

x

Typical saturation profile

layer 1: h1 φ1 k1 ΔS 1 M1

layer 2 : h2 φ2 k2 ΔS 2 M2

layer N: hN φN kN ΔS N MN

P1 P2

L

Water injection

Oil and water production

fluid front position depends on layer properties


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d) Sketch a typical Vertical Equilibrium (VE) displacement situation. Include a typical vertical saturation distribution profile.

Question 5 (10 points) For the two situations below (i and ii) please derive expressions for surface gas production, surface water production, and surface oil production. You may neglect capillary pressures. i) ii)

Current WOC Water

saturation

Depth

Typical VE saturation distribution


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solution: i) (4 points) Oil in stock-tank: QoR / Bo Surface volume of gas: QoRRso / Bo Surface volume of water: 0. ii) (6 points) Oil in stock-tank: QoR / Bo

Surface volume of gas: solution gas + free gas = QoR Rso /Bo+krgµoµgkroBg

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Surface volume of water: = QoRkrwµo

µwkroBw

Question 6 (12 points)

a) List all steps and formulas/equations/definitions used in the derivation of a (one-phase) fluid flow equation.

• Continuity equation • Darcy’s equation (velocity as function of pressure) • Fluid description (density as function of pressure) • Pore description (porosity as function of pressure)

Which coordinate systems are used for the following flow equations?

b)

∂ 2P∂x2 = (

φµck

)∂P∂t

Cartesian (linear)

c)

1r∂∂r

r∂P∂r

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = (

φµck

)∂P∂t

Cylindrical (radial)

d)

1r2

∂∂r

r2 ∂P∂r

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = (

φµck

)∂P∂t

Spherical

e) Which two main types of boundary conditions are normally used to represent reservoir fluid production and injection?

• Bottom hole pressure specified • Production rate specified

f) Write the steady-state form of equation d) above, and solve for pressure as a function of radius for boundary conditions P(r = re) = Pe and P( r = rw ) = Pw


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∂∂r

r2 ∂P∂r

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = 0 ⇒ r2 ∂P

∂r= A ⇒

∂P∂r

=Ar2

⇒ P = −Ar

+ B

P(r = re) = Pe = −Are

+ B

P(r = rw ) = Pw = −Arw

+ B

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

A =Pe − Pw

1rw

−1re

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

, B = Pe −Pe − Pw

1rw−

1re

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ re

P = Pe −Pe − Pw

1rw

−1re

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1r−

1re

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Question 7 (22 points) The general form of the Material Balance Equation may be written as (se attached definitions of the symbols used):

N p Bo2 + Rp − Rso2( )Bg2[ ] + Wp Bw 2 =

N Bo2 − Bo1( ) + Rso1 − Rso2( )Bg2 + mBo1

Bg2

Bg1

− 1⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ − 1+ m( )Bo1

Cr + Cw Sw1

1− Sw1

P2 − P1( )⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

+ Wi + We( )Bw 2 + GiBg2

a) What is the primary assumption behind the use of the Material Balance Equation, and which

"driving mechanisms" or "energies" are included in the equation? solution--- 4 p --- Primary assumption: Zero-dimensional system

(homogeneous system/no flow inside reservoir) Driving mechanisms: -Expansion/contraction of reservoir fluids (including gas cap) -Expansion/contraction of reservoir rock -Aquifer influx -Gas/water injection b) Reduce the equation and find the expression for oil recovery factor ( N p / N ) for the following

reservoir system: • The reservoir is originally 100% saturated with oil at a pressure higher than the bubble point

pressure • The production stream consists of oil and gas • No injection of fluids • No aquifer

solution--- 4 p ---


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Bg2

Bg1

− 1⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ − 1+ m( )Bo1

Cr + Cw Sw1

1− Sw1

P2 − P1( )⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


�

⇒ N p Bo2 + Rp − Rso2( )Bg 2[ ] = N Bo2 − Bo1( ) + Rso1 − Rso2( )Bg2 − Bo1Cr P2 − P1( )[ ]RF =

N p

N=

Bo2 − Bo1( ) + Rso1 − Rso2( )Bg2 − Bo1Cr P2 − P1( )Bo2 + Rp − Rso2( )Bg 2

c) Simplify the expression in b) for the following situations:

i) P2 ≥ Pbp ii) P2 < Pbp, cr and cw may be neglected

solution--- 2 p --- P2 ≥ Pbp

�

N p Bo2 + Rp − Rso2( )Bg2[ ] = N Bo2 − Bo1( ) + Rso1 − Rso2( )Bg 2 − Bo1Cr P2 − P1( )[ ]

�

⇒ RF =N p

N=Bo1Bo2

Bo2Bo1

−1⎛ ⎝ ⎜ ⎞

⎠ ⎟ − Cr P2 − P1( )⎡

⎣ ⎢ ⎤

⎦ ⎥

solution--- 2 p --- P2 < Pbp, cr and cw may be neglected

�

N p Bo2 + Rp − Rso2( )Bg2[ ] = N Bo2 − Bo1( ) + Rso1 − Rso2( )Bg 2 − Bo1Cr P2 − P1( )[ ]⇒ RF =

N p

N=

Bo2 − Bo1( ) + Rso1 − Rso2( )Bg 2Bo2 + Rp − Rso2( )Bg2[ ]

d) Make the following sketches for the reservoir in b):

• A typical curve for GOR vs. time for the reservoir . Explain details of the curve. solution--- 3 p ---

GOR

time

�

P > PbpSg = 0

�

Rso(P > Pbp)

�

Sg < Sgc

�

P < PbpSg > 0


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• A typical curve for oil recovery factor, N p / N , vs. cumulative gas-oil ratio,

�

Rp . Explain details of the curve.

solution--- 4 p ---

�

RF =N p

N=

Bo2 − Bo1( ) + Rso1 − Rso2( )Bg2 − Bo1Cr P2 − P1( )Bo2 + Rp − Rso2( )Bg 2

⇒ RF =A

B + Rp (for a given set of P1 and P2 )

e) Reduce the equation for the following reservoir system:

• The reservoir is originally at bubble point pressure and has a gas cap • The production stream consists of oil and gas • No injection of fluids • No aquifer

solution--- 2 p ---



Bg2

Bg1

− 1⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ − 1+ m( )Bo1

Cr + Cw Sw1

1− Sw1

P2 − P1( )⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


�

⇒ N p Bo2 + Rp − Rso2( )Bg 2[ ] =

N Bo2 − Bo1( ) + Rso1 − Rso2( )Bg2 + mBo1Bg 2Bg1

−1⎛

⎝ ⎜

⎞

⎠ ⎟ − 1 + m( )Bo1

Cr + CwSw11− Sw1

P2 − P1( )⎡

⎣ ⎢

⎤

⎦ ⎥

RF

Rp

�

RF@ Rp = Rso1

�

RF =A

B + Rp


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f) Make the following sketches: • A typical curve for reservoir pressure vs. time for a large gas cap. • A typical curve for reservoir pressure vs. time for a small gas cap.

solution--- 2 p ---

g) Reduce the equation for the following reservoir system:

• The reservoir is originally at a pressure higher than the bubble point pressure and contains oil and water

• The production stream consists of oil, water and gas • No injection of fluids • Water flows into the reservoir from an aquifer.

solution--- 2 p ---



Bg2

Bg1

− 1⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ − 1+ m( )Bo1

Cr + Cw Sw1

1− Sw1

P2 − P1( )⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


⇒ N p Bo2 + Rp − Rso2( )Bg 2

⎡⎣⎢

⎤⎦⎥+Wp Bw2 = N Bo2− Bo1( )+ Rso1− Rso2( )Bg 2− Bo1

Cr + CwSw1

1− Sw1

P2− P1( )+WeBw2

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

P

time

Large gas cap

Small gas cap


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h) Make the following sketches: • A typical curve for reservoir pressure vs. time for a reservoir with a strong aquifer. • A typical curve for reservoir pressure vs. time for a reservoir with a weak aquifer.

Please indicate a (relative) point where the bubble point pressure is crossed. solution--- 2 p ---

Question 8 (6 points) Applying Dietz' stability analysis to displacement of oil by water or by gas in an inclined layer (angle α), we may derive the following formula for the angle (β) between the fluid interface and the layer:

tan(β ) = tan(α ) +

1− Me

MeN ge cos(α )

where the gravity number is defined as

N ge =

( ′ k ro / µo )kAΔγqinj

and Me is the end-point mobility ratio, both computed using endpoint relative permeabilities. a) What is the criterion for the stability of the fluid front?

• solution β > 0 b) When is the front completely stable (in the equation above)?

• solution Me ≤ 1 c) When is the front conditionally stable?

• solution Me > 1

P

time

Strong aquifer

Weak aquifer P=Pb

p

P=Pb

p


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Attachment - Definition of symbols

Bg Formation volume factor for gas (res.vol./st.vol.) Bo Formation volume factor for oil (res.vol./st.vol.) Bw Formation volume factor for water (res.vol./st.vol.) Cr Pore compressibility (pressure-1) Cw Water compressibility (pressure-1) ΔP P P2 1− Gi Cumulative gas injected (st.vol.) GOR Producing gas-oil ratio (st.vol./st.vol.) Gp Cumulative gas produced (st.vol.)

k Absolute permeability

kro Relative permeability to oil

krw Relative permeability to oil

krg Relative permeability to oil m Initial gas cap size (res.vol. of gas cap)/(res.vol. of oil zone) Me End point mobility ratio N Original oil in place (st.vol.)

N ge Gravity number Np Cumulative oil produced (st.vol.)

P Pressure

Pcow Capillary pressure between oil and water

Pcog Capillary pressure between oil and gas

qinj Injection rate (res.vol./time) Rp Cumulative producing gas-oil ratio (st.vol./st.vol) = G Np p/ Rso Solution gas-oil ratio (st.vol. gas/st.vol. oil) Sg Gas saturation So Oil saturation Sw Water saturation T Temperature Vb Bulk volume (res.vol.) Vp Pore volume (res.vol.) WC Producing water cut (st.vol./st.vol.) We Cumulative aquifer influx (st.vol.) Wi Cumulative water injected (st.vol.) Wp Cumulative water produced (st.vol.) ρ Density (mass/vol.) φ Porosity

µg Gas viscosity

µo Oil viscosity

µw Water viscosity γ Hydrostatic pressure gradient (pressure/distance)

exam 2011

Documents

gas compressibility

reservoir gas density

oil compressibility

effective reservoir

nzrt h reservoir oil

compressibility v

gas cap

real gas law p