1

Department of Petroleum Engineering and Applied Geophysics

SOLUTION

Examination paper for TPG4150 Reservoir

Recovery Techniques

Academic contact during examination: Jon Kleppe

Phone: 91897300/73594925

Examination date: December 10, 2014

Examination time (from-to): 0900-1300
Permitted examination support material: D/No printed or hand-written
support material is allowed. A specific basic calculator is allowed.

Other information:

Language: English
Number of pages: 15
Number of pages enclosed: 0

Checked by:

____________________________
Date
Signature
TPG4150 Reservoir Recovery Techniques 2
Final exam December 10, 2014

Symbols used are defined in the enclosed table

Question 1 (5 points)

This question relates to the group project work.

a) State if you participated in a Gullfaks I1 or K1/K2 project

b) List the main production mechanisms in the field that you studied.
c) What was the value used for the gas cap parameter m in your calculations and explain how did
you decide to use that value?
d) How did you obtain Bo and Rso for your calculations?
e) List at least two of the main conclusions of the project.
f) List at least three of the main uncertainties in the material balance calculations.

SOLUTION

Her må de bare vise at de har jobbet med prosjektet

Question 2 (16 points)

For displacement of oil by water in a reservoir cross-section, answer following questions:

a) What does the term "segregated flow" mean, and which factors determine this flow condition?
b) What does the term "diffuse flow" mean, and which factors determine this flow condition?
c) What does the term "vertical equilibrium" mean in reservoir analysis, and when is it a realistic
assumption?
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?
f) What is the Dykstra-Parsons method used for, and which assumptions are made for the
method?
g) What is the Buckley-Leverett method used for, and which assumptions are made for the
method?
h) What is the Dietz method used for, and which assumptions are made for the method?

SOLUTION

For displacement of oil by water in a reservoir cross-section, answer following questions:

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
δP
ie. gΔρ >>
δx
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
TPG4150 Reservoir Recovery Techniques 3
Final exam December 10, 2014

δP
ie. >> gΔ ρ (leads to uniform saturation distribution vertically)
δx
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
∂P
ie. gΔρ >> (the “ultimate” segregated flow)
∂x
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.

Diffuse flow Segregated flow

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
• Constant pressure drop for all layers
• piston displacement
• capillary pressure negligible
g) What is the Buckley-Leverett method used for, and which assumptions are made for the
method?
• Displacement calculations under diffuce flow conditions
• Assumptions
• diffuse flow conditions
• no capillary dispursion at front
• incompressible fluids
h) What is the Dietz method used for, and which assumptions are made for the method?
• stable displacement in inclined systems
• Assumptions
• vertical equilibrium
TPG4150 Reservoir Recovery Techniques 4
Final exam December 10, 2014

• piston displacement
• negligible capillary pressure

Question 3 (15 points)

Write or derive an expression (equation or text) that defines each of the following terms (see list of
symbols at the back):

solution---15x1 point---
(res.vol.)
a) Formation volume factor B=
(st.vol.)
(st.vol. gas)
b) Solution gas-oil ratio Rs o =
(st.vol. oil)
1 ∂V f
c) Fluid compressibility cf = − ( )T
Vf ∂ P
1 ∂φ
d) Pore compressibility cr = ( )T
φ ∂P
e) Total reservoir compressibility cT = cr + ∑ ci Si
i = o,w, g

f) Expansion volume (approximate) due to compressibility

ΔV = V2 − V1 ≈ −V1c(P2 − P1 )
g) Real gas law for hydrocarbon gas PV = nZRT
ρoS + ρgS Rso
h) Reservoir oil density ρoR =
Bo
ρgS
i) Reservoir gas density ρgR =
Bg
ρ
j) Reservoir water density ρwR = wS
Bw
k) Relationship between oil compressibility (undersaturated) and formation volume factor
1 dVo
Co = − , and Vo = VoS Bo
Vo dP
1 dBo
Thus, Co = −
Bo dP
l) An expression for gas compressibility using the real gas law
1 dVg
Cg = − , and PVg = nZRT
Vg dP
1 1 dZ
Thus, Cg = −
P Z dP
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 while microscopic 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?
TPG4150 Reservoir Recovery Techniques 5
Final exam December 10, 2014

• By reducing interfacial tension between rock and fluids, eg. by surfactant additions
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 4 (8 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)

SOLUTION

i) (4 points) Oil in stock-tank: 1/ Bo

Surface volume of gas: Rso / Bo
Surface volume of water: 0.
ii) (6 points) Oil in stock-tank: 1/ Bo
krg µo
Surface volume of gas: solution gas + free gas = R so /Bo +
µg kro Bg
Surface volume of water: = 0.

Question 5 (8 points)

An oil reservoir with an initial gas cap and immobile water is being produced from initial
conditions and down to a final closing pressure. The production stream consists of oil and gas.
TPG4150 Reservoir Recovery Techniques 6
Final exam December 10, 2014

Write (derive) the material balance equations needed, and find an expression for the oil recovery
factor. Neglect water and rock compressibilities.

SOLUTION

* Det er også OK hvis de har begynt med den komplette MBE-ligningen (forutsatt at de husker
den riktig) og reduserer den til denne oppgaven.

Utledning:

General mass balance equation:

⎛ Amount of fluids present ⎞ ⎛ Amount of ⎞ ⎛ Amount of fluids remaining⎞
⎜ in the reservoir initially ⎟ − ⎜ fluids produced ⎟ = ⎜ in the reservoir finally ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ (st. vol.) ⎠ ⎝ (st. vol.) ⎠ ⎝ (st. vol.) ⎠
Oil eq. N - Np = VpSo2/Bo2
Gas eq. mNBo1/Bg1 +NRso1 - Gp = VpSg2/Bg2 + VpSo2 Rso2/Bo2
Water eq. Water and rock are imcompressible, thus Sw1=Sw2=constant
Pores Rock is incompressible, thus Vp is constant
Rearrange equations and add saturations: So2+Sg2+Sw2=1.0
Rearranging yelds:
⎡ mBo1 B B ⎤
⎢ + Rso1 − (1 + m) o1 − (Rso2 − o2 )⎥
Bg1 Bg2 Bg2 ⎦
Recovery factor: RF = ⎣
⎡ Bo2 ⎤
⎢ Rp − (Rso2 − )⎥
⎣ Bg2 ⎦
TPG4150 Reservoir Recovery Techniques 7
Final exam December 10, 2014

⎡ Bg2 ⎤
⎢(Bo2 − Bo1 ) + Bg2 (Rso1 − Rso2 ) + mBo1 ( −1)⎥
Bg1
or RF = ⎣ ⎦
⎡⎣ Bg2 (Rp − Rso2 ) + Bo2 ⎤⎦

Question 6 (8 points)

For the homogeneous reservoir section below

a) Sketch typical capillary pressure curves used for equilibrium calculations of initial saturations.
Label important points.
b) Sketch typical initial water, oil and gas pressures vs. depth. Label important points used and
explain briefly the procedure used.
c) Sketch the corresponding initial water, oil and gas saturation distributions determined by
equilibrium calculations and capillary pressure curves. Label important points and explain
briefly the procedure used.
d) Explain the concepts of WOC contact and free surface, using a sketch

SOLUTION

At the WOC Po-Pw=Pdow, and at GOC Pg-Po. Initial pressures are computed using densities and
assuming equilibrium. At WOC Sw=1,0. At any z value, Pcow is computed from the difference in
Po and Pw, and the corresponding Sw is found from the Pcow-curve. At GOC Sg=0. At any z-
value above the corresponding Sg is found from the Pcog-curve
TPG4150 Reservoir Recovery Techniques 8
Final exam December 10, 2014

• Explain the concepts of WOC contact and free surface, using a sketch

Question 7 (12 points)

a) Start with Darcy’s equations for displacement of oil by water in an inclined layer at an angle
α (positive upwards):
kk A ⎛ ∂P ⎞
qo = − ro ⎜ o + ρ o g sin α ⎟
µ o ⎝ ∂x ⎠
kk rw A ⎛ ∂ ( Po − Pc ) ⎞
qw = − ⎜ + ρ w g sin α ⎟
µw ⎝ ∂x ⎠
and derive the expression for water fraction flowing, f w , inclusive capillary pressure and
gravity.
b) Make typical sketches for water fraction flowing, f w , vs. water saturation, assuming capillary
pressure and gravity may be neglected, for the following cases:
• a high mobility ratio
• a low mobility ratio
• for piston displacement
c) Make a typical sketch for water saturation vs. x for water displacement of oil in a horizontal
system (Buckley-Leverett), assuming capillary pressure and gravity may be neglected, for the
following cases:
• a high mobility ratio
• a low mobility ratio
• for piston displacement

SOLUTION
a) Rewriting the equations as

µo ∂P
−qo = o + ρo gsin α
kkro A ∂ x
µ ∂P ∂P
−qw w = o − c + ρw gsin α
kkrw A ∂ x ∂ x
TPG4150 Reservoir Recovery Techniques 9
Final exam December 10, 2014

and then subtracting the first equation from the second one, we get
1 ⎛ µ µ ⎞ ∂P
− ⎜ qw w − qo o ⎟ = − c + Δρ gsin α
kA ⎝ krw kro ⎠ ∂x
Substituting for
q = qw + qo
qw
fw =
q

and solving for the fraction of water flowing, we get the following expression:
kk A ⎛ ∂ P ⎞
1 + ro ⎜ c − Δρ gsin α ⎟
⎝
q µo ∂ x ⎠
fw =
k µ
1 + ro w
µo krw
b)
“High” mobility ratio

“Low” mobility ratio

Piston displacement
fw

Swir Sor
Sw
c)

} Piston displacement

“Low” mobility ratio

Sw
“High” mobility ratio

}
x
TPG4150 Reservoir Recovery Techniques 10
Final exam December 10, 2014

Question 8 (10 points)

For water displacement of oil in a fractured reservoir the wetting conditions of the reservoir rock
may greatly influence the recovery process.

a) Sketch the following capillary pressure curves:

• A typical imbibition curve for a 100% water wetted system
• A typical imbibition curve for a system that is partly oil wet.
Mark the following items on the curves:
• End point saturations
• The area for spontaneous imbibition
• The area for forced imbibition
b) What is the final (theoretical) oil recovery factor for a 100% water-wet fractured reservoir
under water flooding? Write the appropriate expression.
c) The vertical continuity (contact) between matrix blocks in the reservoir may in some cases
influence significantly the oil recovery. Explain shortly for which situations this is true for the
following processes:
• Water displacement
• Gas displacement

SOLUTION

a)
• A typical imbibition curve for a 100% water wetted system

Pcow

Sw
Swir 1 − Sor

• A typical imbibition curve for a system that is partly oil wet.

Pcow

1 − Sor
Sw
Swir 1 − S s
or
TPG4150 Reservoir Recovery Techniques 11
Final exam December 10, 2014

b)
OIPinitially − OIPfinally Vp ⎡⎣(1 − Swir ) − ( Sor )⎤⎦ 1 − Swir − Sor
RF = = =
OIPinitially Vp (1 − Swir ) 1 − Swir

c)
• Water displacement

a. For a 100% water-wet reservoir, there is no influence of vertical contact on oil

recovery, since all movable oil is recovered by spontaneous imbibition
b. For mixed-wet reservoir, spontaneous imbibition recovers oil only until Pcow = 0
ie. until So = Sors . Thereafter, oil is recovered by forced imbibition by gravity for
Pcow < 0 . The taller the block, the higher recovery. Capillary continuity between
blocks will have the same effect as taller blocks
• Gas displacement

Since the process is a drainage process, for gas to enter the matrix blocks, and thus
replace oil, the difference in phase pressures must exceed the displacement capillary
pressure, ie. Pg − Po > Pcogd . Thus, oil is recovered by means of gravity forces. The
taller the block, the higher recovery. Capillary continuity between blocks will have
the same effect as taller block.s

Question 9 (8 points)

Start with Darcy´s equations for oil and gas (neglect capillary pressure), and

a) Derive an expression for GOR (gas-oil ratio) at surface conditions for a well that perforates
one layer in a horizontal, undersaturated reservoir.
b) Derive an expression for GOR (gas-oil ratio) at surface for a well that perforates one layer in a
horizontal, saturated reservoir (Neglect capillary pressure).
c) Sketch a typical curve of GOR vs. time for an initially undersaturated oil reservoir that is
produced through pressure depletion. Explain all details.

Producing GOR of a well is 1100 (sm3 gas/sm3 oil) and the solution gas-oil ratio (Rso) is 100 (sm3
gas/sm3 oil). Formation-volume factors for oil and gas are: Bo = 2 and Bg = 0,005 .

d) What is GOR at reservoir conditions (rm3 gas/rm3 oil)?

e) What is the fraction of the surface-GOR (sm3 gas/sm3 oil) coming from the free gas in the
reservoir?
f) What is the gas-oil mobility ratio in the reservoir?
TPG4150 Reservoir Recovery Techniques 12
Final exam December 10, 2014

SOLUTION
a)
k A ∂P
qo = − o
µo Bo ∂r
ko A ∂P kg A ∂P k A ∂P
qg = −Rso − = −Rso o
µo Bo ∂r µ g ∂r µo Bo ∂r

qg
GOR = = Rso
qo
b)
ko A ∂P
qo = −
µo Bo ∂r
ko A ∂P kg A ∂P
qg = −Rso −
µo Bo ∂r µ g Bg ∂r
ko A ∂P kg A ∂P
−Rso −
qg µo Bo ∂r µ g Bg ∂r
GOR = =
qo k A ∂P
− o
µo Bo ∂r
kµB
GOR = Rso + g o o
µ g ko Bg
c)
GOR

Rso (P > Pbp )

Sg < Sgc
time
P > Pbp P < Pbp
Sg = 0 Sg > 0

d)

⎛q ⎞
GORs = ⎜ g ⎟
⎝ qo ⎠ s
⎛q ⎞ (q − qos Rso )Bg B 0, 005
GORr = ⎜ g ⎟ = gs = (GORs − Rso ) g = 1000 = 2, 5
⎝ qo ⎠ r qos Bo Bo 2
TPG4150 Reservoir Recovery Techniques 13
Final exam December 10, 2014

e)
kg µo Bo
GORtotal = Rso + = 1100
µ g ko Bg
GOR free gas = 1100 − Rso = 1000 sm 3 gas / sm 3 oil
f)
kg µo Bo
GORtotal = GORsolution gas + GOR free gas = Rso + = 1100
µ g ko Bg
kg µo Bo kµ B 0, 005
GOR free gas = = 1000 ⇒ M go = g o = 1000 g = 1000 = 2, 5
µ g ko Bg µ g ko Bo 2

Question 10 (10 points)

a) List all steps and show formulas/equations/definitions used in the derivation of a one-
dimensional (x) one-phase (oil), horizontal fluid flow equation.

b) Show all details in the derivation of the following equation:

∂ ⎛ k ∂P⎞ ∂ ⎛ φ ⎞
= ⎜ ⎟
∂ x ⎜⎝ µ B ∂ x ⎟⎠ ∂ t ⎝ B ⎠

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

SOLUTION
a)
List all steps and formulas/equations/definitions used in the derivation of a one-dimensional (x)
one-phase (oil) with Black Oil fluid description, horizontal fluid flow equation.
∂ ∂
• Continuity equation: − ( Aρouo ) = ( Aφρo )
∂x ∂t
k ⎛ ∂P ⎞
• Darcy’s equation: uo = − ⎜ o ⎟
µo ⎝ ∂ x ⎠
ρos + ρgs Rso constant
• Fluid description ρo = =
Bo Bo
1 ⎛ ∂φ ⎞
• Pore description: cr =
φ ⎜⎝ ∂ Po ⎟⎠ T

b)
Continuity equation:
∂ ∂
− ( Aρouo ) = ( Aφρo )
∂x ∂t
For constant cross sectional area, the continuity equation simplifies to:
∂ ∂
− ( ρouo ) = (φρo )
∂x ∂t
TPG4150 Reservoir Recovery Techniques 14
Final exam December 10, 2014

Darcy's equation
k ⎛ ∂P ⎞
uo = − ⎜ o ⎟
µo ⎝ ∂ x ⎠
Rock compressibility
1 dφ
cr =
φ dPo
Fluid compressibility
constant
ρo =
Bo
Substituting Darcy´s equation and oil density into the continuity equation:
∂ ⎛ k ∂ Po ⎞ ∂ φ
= ( )
∂ x ⎜⎝ µo Bo ∂ x ⎟⎠ ∂ t Bo
The right hand side (RHS) of the equation may be expanded as:
∂ φ d 1 ∂ Po 1 dφ ∂ Po
( )=φ ( ) +
∂ t Bo dPo Bo ∂ t Bo dPo ∂ t
The final equation then becomes:

∂ ⎛ k ∂ Po ⎞ ⎡c ∂ ⎤ ∂P
⎜ ⎟ =φ⎢ r + (1 / Bo )⎥ o
∂ x ⎝ µo Bo ∂ x ⎠ ⎣ Bo ∂ Po ⎦ ∂t
c)
• Bottom hole pressure specified
• Production rate specified
TPG4150 Reservoir Recovery Techniques 15
Final exam December 10, 2014

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 P2 − P1
Gi Cumulative gas injected (st.vol.)
GOR Producing gas-oil ratio (st.vol./st.vol.)
Gp Cumulative gas produced (st.vol.)
k Absolute permeability
k ro Relative permeability to oil
k rw Relative permeability to oil
k rg 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 p / N 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)