rock permeability

7/30/2019 Rock Permeability

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2.4 Permeability of reservoir rock

1 Darcys law and absolute permeability of rock

2 The unit of permeability

3 The conditions of measuring absolute

permeability

4 Applicable condition of Darcy's law

5 Gas Permeability and Effect of Gas Slippageon Permeability

6 Graduation of formation in permeability


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2.4 Permeability of reservoir rock

7 Factors affecting the magnitude of

permeability

8 Determination of permeability9 Flow Systems of Simple Geometry

10 Average permeability of combinationlayers

11 The relationship of permeability andother p p y of rock

g g

propert


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1 Darcys law and absolute permeabilit of

rockIn 1856, Darcy investigated the flow of

water through sand filters .

His experimental apparatus is shownin Fig

Darcy interpreted his observations so

as to yield results essentially as given inequation.


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Q

Asand filter

h1 - h2

L

= K

upstream

pressure

QL

constant absolute permeability

of rockexperimental apparatus

(KA P1 - )


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Definition of p y

Permeability is the ability of fluids to

pass through a porous material.

The unit of permeability is the Darcy

s ) )1 Darcy

1(c 1(atm cm )2

1 Darcy = 0.987 x 10-8 cm2

permeability

3

)1(c


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Here,

Q represents the rate of flow through thecylindrical sand pack of cross-sectional area A

and length L.

P is pressures drop across the cylindricalsand pack .

is viscosity of the fluid

K, a constant of proportionality, is found to becharacteristic of the sand pack.

p p


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2 The unit of permeability

Darcy CGS PERM SI

L

A

P

Q

K

cm cm ft cm

cm2 cm2 ft2 cm2

atm dyn/cm2 psia MPa

cp p cp mpa.s

cm3/s cm3/s bbl/day cm3/s

darcy cm2 perm m2


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Physical significance of one Darcy

One Darcy is defined as the permeability that

will permit a fluid of 1 centipoise (cp)viscosity

to flowat a rate of 1 cm3/s through a cross

sectional area of 1 cm2 when the pressure

gradient is 1 atm/cm.

1 Darcy (d)=1000 millidarcy (md)


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3 The conditions of measuring absolute

permeability

The porous medium must be 100 per centsaturated with the flowing fluid.

No interaction between the fluid and theporous media

Laminar flow (.

p p


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EXAMPLE 4

If the length of a core sample is 3cm ,and the

cross-section area is 2cm2. Under a pressure

difference of 2atm, the core sample will be

permit a fluid of 1 cp (centipoise ) viscosity to

flow at a rate of 0.5cm

3

/s through.Calculate permeability of core sample .

SOLUTION:

QL 0 5 1 3= = =0.375 D

U d

0.51


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Ror the same core, if the core sample will

be permit a fluid of 3 cp (centipoise ) viscosityto flow at a rate of 0.167cm3/s through,

Under the same pressure difference.

Calculate permeability of core sample .

QLK= =

AP 22=0.375(D) 0


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Characteristic of absolute permeability:

The permeability is a property of rock

alone,dependent on the porous media size and

distribution of throat) only, not on the fluid

used.


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4 Applicable condition of Darcy's law

1) Mid-high permeability

vc =

2) Low permeability

Fig.1 relations between velocity andpressure gradient

s law

3 23.5u


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5 Gas Permeability and Effect of Gas

Slippage on Permeability

1). Calculation of gas permeability

QuL

K =

Fig.2 Pressure distribution in core of linear flow

K= =A(P1 +P2)(P1 -P2) A(P1

2-P2

2)

2P0Q uL 2P0Q uL0 0


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2). Klinkenberg Effect

Klinkenberg Effect(Effect of Gas Slippage )

expressed that the flow rate of gas at solid

surface of pore is not zero.


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The reason of why gas permeability

is greater than liquid Permeability

It has been found that gas permeabilities

sometimes vary with the pressure of gas,

and that gas permeability is greater than

liquid Permeability. The variation is due to

the effect of gas slippage that occurs in

capillary tubes.p y


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3). Effect of gas slippage on Permeability

Measurements

the reciprocal of the mean

1000

80

0

600

400

0 2 4 6 8 10

1/p,1/MP

1

2

3


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() of the mean pressure of the t t

The date obtained with the lowest-molecular-weightgas yield the straight line with the greatest slope

,indicative of a greater slippage effect.

All the lines when extrapolated to infinite mean

For each gas a straight line is obtained for the

observed permeability as a function of the

1pressure ( ) intercept the permeability axis= 0pat a common

This point is designated kL or the equivalent liquid

permeability.

f th f th test.


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k= absolute of Klinkenberg effect correctedpermeability,

b=constant for a given gas in a given medium

Kg=measured permeability using the Darcy equationfor gas flow.

The linear relationship between the observed

permeability and the reciprocal of meanpressure

Kg = K( p

= mean flowing pressure of thep

(1+ b / ))


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Kg = (1+ )

P

1

2d n

Kg

K

4C P

rb = =

b

pK

2


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> 1 >1000 Very good

10.1 1000100 good

0 1 0 01 10010 middle

0.010.001 101 fair

< 0.001 < 1 Non-permeable

formation in Graduation of

permeability

grade

permeability evaluation

10-3um22u

0.10.01

< 1


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7 Factors affecting the magnitude of

permeability(a) Shape and size of sand grains

(b) Grain packing and Cementation

(c) Fracturing and solution

(d) Pore Size and Connectivity

(e) Compaction

P Si d C ti it


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8 Determination of permeability

1) The experimental laboratory determinations

A . determinations on small samples of

extracted media

Apparatus for measuring of permeability with


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Apparatus for measuring of permeability with gas

K = 2Q 0P0uLA(P1

2 - P22 )


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Vertical pipe flow meter method

Bu

L

K= 103


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B . determinations on full-diameter core

samples

2calculating permeability by pore radiusand porosity

2

2

r

8K =

tortuosity of the porous medium


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Here: a.b.c are constants, characteristics

of the fluid and the porous medium

3) Estimating permeability by log

aCK =

b

wS


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9 Flow Systems of Simple Geometry

Horizontal Flow.Horizontal rectilinear(steady-state flow is

common to virtually all measurements of permeability.

kA(P1 -P2 )

LQ =

y p y


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Radial Flow.

A radial-flow system, analogousto flow into a wellbore , is idealized in Fig.

Q

re the radius at which Peis measured

rw the radius at which Pwis measured.

2 kh(P- Pe w

ln e / w


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10 Average permeability of combination

layers

1). Consider the case where the flow system iscomprised of layers of porous rock separated

from each other by infinitely thin impermeable

barriers as shown in Fig. Parallel combination

Linear flow

n

j = 1

n

kj

k =

=

) y

y y p

h

j

j

h


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Radial flow

Qt = Q1+ Q2 + Q3

ht = h1 + h2 + h3

n

j = 1

n

kj

k =

=

+ h + h

h

j

j

h


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2) Another p

systems is to have the beds of different

permeability to be arranged in series as

Linear flowP1- P2 = P1+P2+P3

Qt = Q1= Q2 = Q3

L

Lj / k

k =n

j=1

) possible combination for flow

shown in Fig. series combination

j


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Radial flowP1- P2 = P1+P2+P3

Qt = Q1= Q2 = Q3

Q

K= 1Ki Ri-1

K2h(P- P)e w

u lnRe RR

e

Rw

ln


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11 The relationship of permeability and

other property of rock

1)Bundle of capillary tubes pack

Fig. parallel bundle of capillary tubes pack


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Consider a system comprised of a bundle of

capillary tubes of the same radius and length;

The equation for flow of fluids in circular conduits:

Poiseuilles

r04 (P1 - P2 )

8uL

q

qthe volume rate of flow, cm3/s

(P1-P2)pressure loss over length L,

ro radius of capillary tube , cm,

fluid viscosity,


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2)The relationship of permeability and pore radius

If the fluid-conducting channels in a porous medium

could be represented by a bundle of parallel capillary

tubes of same diameters, then the flow rate through

this system is:

Q =nAr4P

nnumber of capillary tubes per unitcross-

sectional area

rradius of capillary tube


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Equating Darcys and Poiseuilles equation:

K AP

'Q =

From Darcys law it is also known

'

4

Q =

KAP nAr P K=

uL 8uL

4

(1)n

r


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=

Therefore:

By definition, the porosity

nAr2L

AL

VP

Vb(2)

(3)

= = nr2

r2

8

8

K

K= r=


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surface area S is the total area exposedthe pore space per unit of bulk volume .

3)The relationship of permeability andspecific

within

2 2a nA2rL= 2nr= nr2 S=

S =

=

Vb AL

=

r r

2

8K

2 S = =

28K

24 3


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Let Sp be the internal surface area per unit of pore

volume ,Ss be the total area exposed within the

pore space per unit of grain volumea

S =V

P

S = SP = (1 - )SS

aSS = =Vb(1-) 1-

P

S


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EXAMPLE 1

A core sample from a uniform sandstone formation

10-3has a permeability of480

of 0.17

(a) The average pore throat radius of thecore.

m2 and a porosity


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problem

1.What is permeability of rock? What are theconditions that should be met when measuring

absolute permeability of rock?

2.What phenomenon will appear when using gasto measure permeability of reservoir rock? How

does it influence the result of measurement?

3.

rock permeability

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