1

Reservoir Petrophysics PETR2311 Spring 2011

Test 1 – 22nd

February 2011, Thursday (3:30 PM to 4:50 PM)

IMPORTANT:

a) Please come to class 5 minutes before the schedule time and leave your belongings at the

corner of the room as instructed by the proctor.

b) You will be provided 8”×10½” graph paper with a form.

i. Write all the details on that sheet.

ii. All subsequent sheets will be blank - write your student ID on the top left corner.

iii. Write page number on top right corner

c) Use a separate sheet for each question if possible

d) Don’t write on the back side of the paper. Use one side only.

e) You can use either pencil or pen (black or blue) for this test.

f) You can bring a simple scientific calculator to the exam. Programmable calculator,

graphing calculators, computers, tablets, cellphones, PDAs, smartphones are NOT

allowed. If you don’t have a simple scientific calculator, please borrow one from your

friends.

g) Answer sheets and graph paper will be provided during the exam.

h) This is a close book test. No written material, books etc. will be allowed. Please bring

only pencil, pen, eraser, calculator and ruler.

i) Unit conversion table will be provided during the test.

2

UNITS CONVERSION

AREA

1 Acre = 43560 ft2

1 ft2 = 0.0929 m

2

1 hectare = 10000 m2

LENGTH

1 Cm = 0.3937 in

1 Ft = 30.481 cm

1 Ft = 12.00 in

1 Km = 0.6214 mile

PRESSURE

1 Atm = 14.696 psi

1 Bar = 0.987 atm

1 Bar = 0.1 MPa

1 dyne/cm2 = 6.895 kPa

VOLUME

1 Bbl = 42 US gal

1 US gal = 3785.43 cm3

3

PROBLEM 1 (10 points = 2 + 1 + 1 + 2 + 3 + 1)

1) Name the eras in the Geological Time scale starting from the oldest to the newest?

2) To which era do most reservoirs belong?

3) Name the period(s) to which most reservoirs belong.

4) Name the epochs of the newest era from newest to oldest.

5) What are the four types of rocks of most interest to Petroleum Engineers and why is each

important. (I am expecting short answers)

6) What is the definition of the Darcy unit of permeability?

SOLUTION:

1) Precambrian, Paleozoic, Mesozoic, Cenozoic

i. 1 points for getting all names right

ii. 1 point for getting the order right

2) Mesozoic

3) Cretaceous, Jurassic

4) Holocene, Pleistocene, Pliocene, Miocene, Oligocene, Eocene, Paleocene

5) Most interesting rock types for Petroleum Engineers:

i. Sandstone – Reservoir

ii. Shale – Seal, and starting to also be reservoir

iii. Carbonate – Reservoir

iv. Salt – seal and structure

6) Darcy definition

4

PROBLEM 2 (10 points = 3 + 3 + 1 + 1 + 1 + 1)

The figure below is an illustration of body centered cubic (BCC) unit cell. The spheres at the

corners have radius R1 and the one at the center of the unit cell has radius R2.

BA

CD

A1 B1

C1D1

3D representation of a BCC unit cell Wireframe of the Unit Cell

a

a

R1

R2

R1

a

Front View of the Unit Cell Section along plane AA1C1C

a) What is the size of the unit cell in terms of R1 and R2.

b) What is the grain volume

c) What is the porosity

d) What engineering type of porosity

e) If R1 and R2 are very different, we would say that this sandstone has poor sorting. For

such a poorly sorted sandstone, suppose R1 and R2 were both reduced by a factor of 10.

Qualitatively, what effect do you expect on

i. Porosity?

R1

5

ii. Permeability

SOLUTION:

1 1

2 2

1 1

2 2

1 2

1 2

1 2

2

2( ) 2

2( ) 3

2( )

3

AB AD BC DC a

AC AC a

AC AA AC

R R a a

R R a

R Ra

33 1 2

1 2

3 3

1 2

3 33 3

1 21 2

3 3

1 2 1 2

8( )

3 3

18

8

4 4

3 3

4331 1 1

8( ) 2( )

3 3

cell

grain sphere sphere

grain

cell

R RV a

V V V

R R

R R R RV

R RV R R

d) Effective porosity

e) Porosity – no change; Permeability decreases

6

PROBLEM 3 (20 points = 8 + 2 + 5 + 5{1 + 3 + 1})

In porosity experiment of an irregular porous rock core the following measurements were made

Dry sample mass = Md

Mass of the water-saturated sample = Ms

Mass of the saturated sample immersed in an insoluble liquid = Mb

a) Show that the porosity of the sample measured using the above measurements is

independent of the liquid density.

b) What is the engineering classification of this porosity and why? Explain clearly?

c) Uncertainty in Measurements:

i. Derive an expression for uncertainty in the measurement of porosity using the

following measurements

Md = Md,best ± δMd

Ms = Ms,best ± δMs

Mb = Mb,best ± δMb

ii. Enclosed is a set of example measurements for the above masses

a. Calculate the best estimate of the porosity.

b.What is the uncertainty in the porosity these data?

c. Do you think the experiments are consistent? Why? (Tabulate your

results)

SOLUTION

a)

Given:

Dry sample mass, d gM m

Mass of sample + liquid= s g wM m m

Md δMd Ms δMs Mb δMb

25 0.2 26 0.1 21 0.1

26 0.1 27 0.2 22 0.1

24 0.3 25 0.3 20 0.1

23 0.2 24 0.1 19 0.1

7

Mass of saturated sample in liquid = ( )b g w g v wM m m V V

Assume density of liquid = w

Find:

Total porosity, v

g v

V

V V

?

Procedure:

1

Multiply by in the numerator and denominator

1( )

gv

g v g v

w

w g

w g v

VV

V V V V

V

V V

Denominator

( )

( )

( )

b g w g v w

s g v w

g v w s b

M m m V V

M V V

V V M M

Numerator

( )b g w g v w

d v w g w v w

d g w

g w d b

M m m V V

M V V V

M V

V M M

Therefore, porosity is:

1( )

1

w g

w g v

d b

s b

V

V V

M M

M M

b)

8

Effective porosity, since void space is measured by the amount of liquid which enters the void.

Hence voids need to be all connected.

c)

2

2

1

1

d b

s b

d s b

d b

s s b

s d

b s b

M M

M M

M M M

M M

M M M

M M

M M M

Md δMd Ms δMs Mb δMb φ

25 0.2 26 0.1 21 0.1 0.200

26 0.1 27 0.2 22 0.1 0.200

24 0.3 25 0.3 20 0.1 0.200

23 0.2 24 0.1 19 0.1 0.200

1/(Ms-Mb)

Md-Mb Ms-Md ∂φ/∂Md δMd ∂φ/∂Ms δMs ∂φ/∂Mb δMb δφ

0.2 4 1 0.2 0.2 0.16 0.1 0.04 0.1 0.06 0.2 4 1 0.2 0.1 0.16 0.2 0.04 0.1 0.056 0.2 4 1 0.2 0.3 0.16 0.3 0.04 0.1 0.112 0.2 4 1 0.2 0.2 0.16 0.1 0.04 0.1 0.06

Then use the formula:

2

2 2 2 2

1

1

1

1 1 1 1

0.06 0.056 0.112 0.06

i

9

PROBLEM 4 (15 points = 4 + 6 + 5)

Given two pipes of sand with inside diameters of D = 2 inch which contains different sands in

each section. The properties of each section are given in the table below. There is a large tank at

the top of the pipe which is fully filled with liquid and has a height H.

Section A B

Permeability kA = 100 mD kB = 50 mD

Length LA = 8.50 LB = 5.6

Angle from horizontal θA = 90° θB = 125°

B

2.50

1

2

125°

8.50 3

A

4

5.6

a) What is the expression and values in atm for pressure at positions 1, 2, 3 and 4 assuming

there is no flow at all, as if the ends of the pipes were closed.. Assume whole system is at

1 atm.

b) Using these pressures and if the viscosity of the fluid is μ, then what is the expression for

the flowrate of the fluid through the sand filled pipes A and B in terms of D, μ, kA, kB,

…., LA, LB, ….θA, θB, …

c) Using the values in the table and assuming the fluid to be salt water, what is the flow rate

through the pipe. Give proper assumptions or rules of thumb.

SOLUTION:

a)

1

2

3

4

1( )

1( )

1( ) sin

1( )

B B

A

P atm

P atm gH

P atm gH gL

P atm gH gL

10

b)

3 1

2

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

q cm s k D P atm

A cm cP L cm

2 4

2 3

( ) ( )

( ) ( sin )

A A A A A A

A A A A

B B B A B B B

B B A B

q k P k P P k gL

A L L L

P Pq k P k k gL

A L L L

c)

11

PROBLEM 5 (15 points = 3 + 6 + 2 + 3 + 1)

You own a one-person oil company. You make your living by draining very thin reservoirs

using horizontal wells. Here is an example of one of your current projects, seen from two

different views:

You know from your vast experience working in this geological setting that

The reservoir is about 1 km along the well, 1 km wide away from the well, and 10 cm

thick. The well is designed to match one edge of the reservoir.

The porosity of the reservoir rock is about 20%

The oil saturation in the reservoir is about 50%

The viscosity of the oil is about 100 centipoise

The average permeability of the reservoir rock is about 100 millidarcies

The pressure at the point furthest from the well is 14700 psi above atmospheric pressure

In order to work out your financing for the project, you need to answer the following four

questions. Express your answers in cgs units:

A. How much oil is initially in place? (OIIP)

B. What is the total rate of flow of all fluids into the well?

1 km

1

kmk

m

1 km

10 cm

well

Map View, i.e., as

seen from above

3D view

12

C. What is the total rate of flow of oil into the well? Since we have not studied multiphase

flow yet, you may assume all fluids are produced proportional to their saturations.

D. How long will it take to completely drain this reservoir of oil if no further oil migrates

into the reservoir but the pressure remains constant? In addition to cgs units, please provide the

time in years as well.

E. If you seek investors based on the value of the OIIP you estimated in Part A, why might they

not provide as much investment as you would like?

SOLUTION

Given

L = 1 km = 1000 m

W = 1 km = 1000 m

H = 10 cm

Φ = 0.2

Soil=0.5

µ=100 cP

k = 100 mD

ΔP = 14700 psi

A) Find OIIP

7758 oiOIIP Ah S

2

2

3

( ) 7758 ( ) ( )

1( ) 1( )7758 ( ) ( ) ( )

4046.9( ) 30.481( )

1( ) 1( )7758 1000( ) 1000( ) 10( ) 0.20 0.50

4046.9( ) 30.481( )

62892.39( )

0.1590( )62892.39( )

oi

oi

OIIP bbl A ac h ft S

ac ftL m W m h cm S

m cm

ac ftm m cm

m cm

bbl

mbbl

6 3

3

9 3

10 ( )

1( ) 1( )

9.99989 10 ( )

cm

bbl m

cm

B) Find total flow rate

c

q k P

A L

13

23 1

3 1

( ) ( ) ( )( )

( ) ( )

0.1( ) 100000( ) 10( ) 1 14700( ) 1( ) 1

1 1 1( ) 14.696( ) 100000( )

100( )

ck D A cm P atmq cm s

cP L cm

D cm cm psi atm

cP psi cm

cm s

C) Oil flow rate

3 1

3 1

0.5 100( )

50( )

oil oil totalq S q

cm s

cm s

D)

3

3 1

108

8

( )

( )

102 10 ( )

50

2 10 ( ) 1( )6.342

365( ) 24( ) 3600( )

oil

oil

V cmt

q cm s

s

s yryears

days h s

E) STOIIP

14

PROBLEM 6 (10 points = 6 + 4)

A stream complex consists of rocks formed by overlapping streams. Since the water in a stream

channel is relatively fast, the channels often contain rocks consisting of large particles. Between

stream channels, flow only occurs during floods. Flood flow is unconfined and slower, leading

to overbank rocks consisting of smaller particles. Here is a picture of a modern stream complex.

Suppose the vertical cross section shown looks like this:

Assume only two permeabilities are present in the reservoir: a channel permeability and an

overbank (between the channels) permeability. One is 10 mD and the other is 200 mD.

20 m2

60 m2 70 m

2

Seal = 200 m2

m2

Seal = 300 m2

Channels

Overbank reservoir

= 850 m2

15

a) Derive the general expression for average permeability of a reservoir with permeability

variation perpendicular to the flow.

b) If this particular reservoir is produced by flow out of the picture, what average

permeability will be observed?

SOLUTION:

a) Lesson 7 Slide 17

b)

Given:

2

2

2

200 300 500

60 20 70 150

850

0

200

10

seal

channel

overbank

seal

channel

overbank

A m

A m

A m

k mD

k mD

k mD

Procedure

i i

avg

i

k Ak

A

Seals will not be included

200 150 10 850

150 850

3850038.5

1000

channel channel overbank overbankavg

channel overbank

k A k Ak

A A

mD

16

PROBLEM 7 (20 points)

Permeability variation in a deltaic reservoir

Suppose a deltaic reservoir is 90 m thick and has a vertical linear permeability variation with 10

mD permeability at the bottom and 100 mD permeability at the top. If this reservoir is subjected

to a horizontal pressure gradient, what fraction of the flow comes from the top 10 m of the

reservoir. (I suggest you draw a picture.)

SOLUTION:

Z = 0, K = 10

mD

Z = 90 m, K = 100

mD

dz

w

17

2 2

1 1

min min

max min max min

90

0

90

1080

- - - 10( ) : ( ) ( ) 10

- 100 -10 90

( 10) ( )

( 10)

( 10)

( 10)

Q z

Q z

total

top

K K z z K zK z K mD z m

K K z z

K P z PdQ A w dz

L L

w PdQ z dz

L

w PQ z dz

L

w PQ z dz

L

F

90 90 80 80

80 0 0 0 10 90 90 90

0 0 0

802

0 10 90 2

0

( 10) ( 10) ( 10) ( 10) 1

( 10) ( 10) ( 10)

( 10) 0.5 (80) 10(80) (80) 1 1 1

(0.5)(90) 10(90) (9( 10)

top

top

z dz z dz z dz z dzraction

z dz z dz z dz

z dzFraction

z dz

5019%

0) 55