Class 13 (Absolute Permeability) petroleum engg

reservoir

1. Permeability
Gas
Oil
Water
•Most influential parameter in
determining the production
capabilities of a producing
formation
•A flow property i.e. dynamic
in nature
Hydrocarbon phaseDepth
Water
Oil-water
transition
zone
Gas-oil
transition
zoneOil
Gas
Water phase
}
}

2. Void spaces
contributes
to absolute
Porosity
Permeable
spaces
contributes
to effective
Porosity
Connectivity between pores is
important for fluid movement
Permeability

3. Permeability
The ability of a material to transmit a fluid.
It is a measure of how fast the fluid can travel through
the rock or sediment

4. -easy to flow
-high permeability
-difficult to flow
-low permeability
A measure of how easily a fluid can pass through a
porous medium.

5. Commonly, reservoirs contain 2 or 3 fluids
(1) Water-oil systems (2) Oil-gas systems (3) Water-gas
systems (4) Three phase systems (water, oil, and gas)
Permeability
FREE WATER
GRAINGRAIN
BOUND WATER
FREE WATER OIL
OIL
RIM

6. Permeability
Absolutepermeability:isthepermeability
ofaporousmediumsaturatedwitha
singlefluid.
Effective permeability: is a measure of the
conductance of a porous medium for one
fluid phase when the medium is saturated
with more than one fluid.
Relative Permeabilityis the ratioof the
effective permeability of a fluid at a given
saturation to some base permeability
To evaluate multiphase systems, must consider the effectiveand
relativepermeability

7. Absolute Permeability
The permeability of a rock
•measure of its specific flow capacity
•capacity to transmit fluid
•ability to transmit fluid through a rock when
a single fluid or phase is present
•fluid conductivity of a particular porous medium
•represents the reciprocal of the resistance that the
porous medium offers to fluid flow
•is the proportionality constant between the fluid flow
rate and an applied pressure or potential gradient

8. CONNECTED
FRACTURES
Permeability depends on
Pore Connectivity
UNCONNECTED
PORES
Most important rock parameter pertinent to fluid flow
Relates to the presence of fractures and interconnected voids

9. Porosity and Permeability
porous sediment:
< 40% porosity
hard rock:
<1% porosity
porosity: volume proportion made up of voids
permeability: connectedness of voids, dictating capacity to transmit flow

10. Darcy's Law:Darcy,s original experiment of the flow of water through
sand is analogous to the flow of a fluid through a core plug
(h
1-h
2)
Q = KA --------
L
Q = volumetric flow rate through the core plug in m
3
/sec or ft
3
/sec
K = proportional constant also defined as hydraulic conductivity in m/sec or ft/sec
A = cross-sectional area of the core plug in m
2
or ft
2
L = length of the core plug in m or ft
h
1 and h
2represent the hydraulic head at inlet and outlet , respectively in m or ft
In terms of pressure gradient Negative sign as increase of pressure and
dP and length is opposite
Q = -KA---------where dP = hg
dL difference between upstream and
downstream pressure

11. Restriction of Darcy’s investigation
flow of water through sand packs that were 100% saturated by water
More generalized formula for other fluids
k dP
Q = ---A ---------where, k = permeability
dL = viscosity of the fluid
With the following assumption
•the core plug is 100 % saturated with the flowing fluid
•the flowing fluid is incompressible
•the flow is horizontal, steady state and under the laminar flow
•the flow of fluid through the porous medium takes place under viscous regime
•the flowing fluid does not react with the porous medium

12. kdP
Q = ---A ---------where, k = permeability
dL = viscosity of the fluid
The above equation can be integrated when
the geometry of the system through which
fluid flows is known. For the simple linear
system shown in Figure, the integration is
performed as follows:
kA
Q
0
L
dL= ------
p1
p2
dP

k A
Q(L-0) = ------(P
2 -P
1)

Since P
1is greater than P
2, the pressure term can be rearranged and the resulting
Equation will be
kA
Q = ------(P
2 -P
1)
L
kA
Q = -----(P
1 –P
2)
L

13. With a flow rate of one cubic centimeter per second across a cross sectional area of
one square centimeter with a fluid of one centipoise viscosity and a pressure
gradient at one atmosphere per centimeter of length, it is obvious that k is unity.
For the units described above, k has been arbitrarily assigned a unit called Darcy
in honor of the man responsible for the development of the theory of flow through
porous media. Thus, when all other parts of equation above have values of unity, k
has a value of one Darcy.
kA
Q = -----(P
1 –P
2)
L
Q = flow rate in m
3
/sec
k = absolute permeability in m
2
A = cross-sectional area in m
2
P
1-P
2= flowing pressure drop in N/m
2This equation is known as Darcy’s law and is
extensively used in reservoir engineering
calculations for determining the absolute
permeability of a reservoir rock
(Cm
3
/sec)(1.0 X 10
-7
N sce/cm
2
)
1 darcy =----------------------------------------------
(cm
2
)(10.1325 N/cm2)
= 9.869 x 10
-9
cm
2
= 9.869 X 10
-13
m
2
= 1.062 x 10
-11
ft
2
1 D = 1000 mD or 1mD = 0.001 D

14. APPLICATION OF DARCY’S LAW TO
INCLINED FLOW OR DIPPING FLOW
The vertical coordinate or the
gradient should also be
accounted for by calculating
absolute permeability
Q = flow rate
k = absolute permeability
A = cross-sectional area
= fluid viscosity
P1-P2 = flowing pressure drop
= fluid density
g = acceleration due to gravity
= angle of inclination
kP
1-P
2
Q = --A ---------+ gsin
 L

15. APPLICATION OF DARCY’S LAW TO
RADIAL FLOW
k dP
Q = ---A ---------
dL
For radialflow
dL is now replaced with dr
A is now 2rh
Using these quantities the Darcy equation
can be integrated between the well bore
and the external boundary of the system
as follows
r
edr k2hp
e
Q-------= ---------dp
r
wr p
wf
Solving for the flow rate
2kh(P
e–P
wf)
Q = --------------------------
ln(r
e/r
w)

16. k
ih
i
Kavg = ---------i= 1 to n
h
i
PARALLEL FLOW
Vertically stacked porous media with varying
permeability and thickness that are separated
from one another by infinitely thin impermeable
barriers that preclude the possibility of cross flow
or vertical flow
k
1Wh
1P
Flow layer Q
1= -------------
L
k
2Wh
2P
Flow layer Q2 = ---------------
L
k
3Wh
3P
Flow layer Q3 = --------------
L
Q
t= Q
1+ Q
2+ Q
3
h
t = h
1+ h
2+ h
3
k
avgWh
tP
Q
t= -------------
L
k
avgWh
tP
-------------=
L
k
1Wh
1P k
2Wh
2P k
3Wh
3
-----------------+ -----------------+ ------------------
L L L
K
avgh
t= k
1h
1+ k
2h
2+ k
3h
3

17. SERIES FLOW
k
1WhP
1
Flow layer1, Q
1= -----------------
L
1
k
2WhP
2
Flow layer2, Q
2= -----------------
L
2
k
3WhP
3
Flow layer 3, Q
3= -----------------
L
3P
1 = P
1–P
2, P
2= P
2–P
3, P
3= P
3–P
4
P = P
1–P
4= P
1+ P
2+ P
3 andQ
t = Q
1= Q
2 = Q
3
Now, Darcy’s law can be written for the total flow rate as k
avgWhP
Q
t= -----------------
L
Q
t L
----------=
k
avgWh
Q
1 L
1
----------+
k
1Wh
Q
2 L
2
----------+
k
2Wh
Q
3 L
3
----------
k
3Wh
L L
1 L
2 L
3
----------= -----------+ -----------+ ------------
K
avg K
1 K
2 K
3
L
i
Kavg = ---------i= 1 to n
L
i
--------
k
i

18. FACTORS AFFECTING ABSOLUTE PERMEABILITY
Rock related factor
basic characteristics
structures
indigenous properties
grain size, shape
and clay cementing
Fluid phase related factor
the physical and chemical characteristics of the fluid
presence of swelling (montmorilonite) and non swelling (kaolinite, illite) clay
mixing of incompatible water in pore spaces of reservoir rock

19. Mechanical factor
Generally absolute permeability is inversely proportional to overburden pressure
because core samples are compacted due to overburden and fluid flow through
such samples is rather squeezed resulting in a reduction in absolute permeability
Thermodynamic factor
Ideally temperature should not have any effect on the absolute permeability
because varying temperature only affects liquid viscosity
FACTORS AFFECTING ABSOLUTE PERMEABILITY

20. 2
101010101010101010-18 -16 -14 -12 -10 -8 -6 -4 -2
B
PERMEABILITY, cm
1nd 1d 1 md 1 d 1000 d
Clay Silt Sand Gravel
Shale Sandstone
argillaceousLimestone cavernous
Basalt
Crystalline Rocks

21. Laboratory measurement of absolute permeability
Based on direct measurements of individual variables
-Flow rate
-Pressure drop
-Sample dimensions
-Fluid properties
Darcy formula for fluids
dP
Q = -K A ---------
dL
For permeability measurements, the residual fluids or in situ formation fluids
are removed so that the sample is 100% saturated with air.
Absolute permeability can only be measured by conducting a flow experiment
In a porous media using
1) Non-reactive liquids
2) Gases

22. Absolute permeability using liquids1. Formation water/brine
2. Degassed crude oil
STEPS
1. The dimension of the core plug is noted
2. The core plug sample is housed in a
Viton sleeve which in turn is mounted in a
Hassler core holder.
3.Anappropriatenetoverburdenorconfiningpressureisappliedradiallytothe
coreviaahydraulichandpump.Theconfiningpressurealsohelpspreventthe
flowofliquidthroughtheminuteannularspacebetweenthecoreplugandthe
sleeveduringtheflowexperiment,
4.Aconstantreservoirtemperatureismaintainedusingtheclimaticairbath.

23. 6.Thepressuredropacrossthecoreplugis
monitoredusingacomputarizeddatalogging
systemandaconstantorsteadypressure
dropisrecordedforcalculation.
5.Adisplacementpumpandfloatingpistonsamplecylinderisusedtoinitiatethe
flowofbrineordegassedcrudeoilateitherconstantrateorconstantdifferential
pressure.Generallyatconstantflowratewheretheinletpressureismonitoredand
theoutletisnormallyatatmosphericpressure
7.Theflowexperimentisrepeatedsometimesbyvaryingtheliquidflowratesto
determinetheratedependencyontheabsolutepermeability.
8.Theviscosityofthebrineortheoilismeasuredatthefloodingpressureand
temperatureconditionsifunknownfromothersources.
9.TheabsolutepermeabilityofthecoreplugsampleisdeterminedusingtheDarcy
equation

24. Absolute permeability using GasesCommonly used gases
1.Nitrogen
2.Helium
3.Air
Advantages
1.Clean
2.Non-reactive
3.Does not influence pore network
Modification of experimental procedure
1.Gas valve should be opened
2.Use of constant differential pressure
Modification of Darcy equation for
calculation of permeability using gases
•In case of incompressible fluid flow, the
flux (Q/A) is constant at all section along the
flow path.
•In case of gas (compressible), the pressure
drop along the flow path is accompanied by
gas expansion that also results in an increase
in the flux

25. Equation for determination of absolute permeability
Theproductofflowratesandpressuresincaseofinletandoutletpointare
equatedbyusingBoyle’slawconsideringtemperatureisconstant
Q
1P
1=Q
2P
2
Q
1P
1=Q
2P
2=Q
avgP
avgQ
avg=avg.gasflowrate
P
avg=avg.pressure
The Darcy equation can be expressed in terms of average gas flow rate
kA(P
1-P
2)
Q
avg = --------------
μL
Theflowrateofgasisnormallymeasuredattheoutletofthecoreplug,Q
2
Q
2P
2 kA(P
1-P
2)
---------------= --------------
(P
1+P
2)/2 μL
kA(P
2
1-P
2
2)
Q
2= --------------equation used for determination of absolute
2μLP
2 permeability of core plug samples using gases

26. Klinkenberg Effect
The higher permeability value obtained in comparison to the liquid
flow for the same core sample –why?
Klinkenberg’sobservationsisthatthesamecoresampleshowedatrendof
increasingpermeabilityasafunctionofincreasingreciprocalmeanpressure
{1/[(P
1+P
2/2]}whenhydrogen,nitrogenandcarbondioxidewereused.These
variationwereascribedtoaphenomenoncalledGASSLIPPAGEthatoccurs
whenthediameterofthecapillaryopeningsapproachtothemeanfreepathofthe
gas.Themeanfreepathofthegasisafunctionofitsmolecularsizeandkinetic
energy.
Mathematical expression
1
k
gas= k
liquid+ m ------
P
mean
K
gas= measured gas permeability
k
liquid= eq. liquid permeability
m = slope of the straight line fit
P
mean= (P
1+P
2)/2

27. ABSOLUTE PERMEABILITY
1.In an experiment similar to that of Darcy’s, the flow rate of water was observed
to be 5.0 cm
3
/min. If the experiment were repeated with oil, what would be the
flow rate for oil ? The difference between the upstream and downstream
hydraulic
gradient h are the same for both the experiments.
Additional data:
Oil viscosity = 2.5 cP, oil density = 0.85 gm/cm
3
Water viscosity = 0.8 cP, water density = 1.0 gm/cm
3
More generalized formula for fluids
k dP
Q = ---A ---------where, k = permeability
 dL = viscosity of the fluid

28. 2. Brine flood in a 1.9 in-long and 1.5 in-diameter core plug from the North Sea
resulted in a stabilized pressure drop of 46.05 psi. The flood was carried out
at 0.05 mL/min with brine viscosity of 0.443 cP. Determine the absolute
permeability of this plug in millidarcies.
3. Three beds of equal cross section have permeabilitiesof 100, 200 and 300 mD
and lengths of 50, 15 and 85 ft, respectively. What is the absolute permeability
of the beds placed in series?
L
i
Kavg = ---------i= 1 to n
L
i
--------
k
i

29. 4. The beds of 50, 110 and 795 mD, and 7, 7 and 15 ft thick respectively, are
conducting fluid in parallel flow. If all are of equal length and width, what is
the average permeability?
k
ih
i
Kavg = ---------i= 1 to n
h
i

30. 5. Following data were obtained during a nitrogen flood in a 1.5 cm diameter
and 3.0 cm long core plug sample. Determine the Klinkenberg corrected
absolute permeability of the core. Nitrogen viscosity 
g= 0.02 cP,
downstream pressure (P
2) is maintained atmospheric.
Run
Number
Q
g(cm3/s)Upstream pressure, P
1
(atm)
1 5.11 1.95
2 18.15 2.45
3 35.61 3.11
4 62.31 3.55