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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.
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