permeability
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PERMEABILITY
Definition (ABW, Ref: API 27)
◦ … permeability is a property of the porous medium and is a measure of the capacity of the medium to transmit fluids
◦ … permeability is the fluid conductance capacity of a rock, or it’s the a measure of the ease with which the rock will permit the passage of fluids.
Reservoir Rocks and Fluid properties
Permeability
Reservoir Rocks and Fluid properties
Permeability theory
Permeability is an INTENSIVE property of a porous medium (e.g. reservoir rock)
Permeability
Reservoir Rocks and Fluid properties
Permeability
Permeability which will permit flow of one centipoise fluid to flow at linear velocity of one cm per second under a pressure gradient of one atmosphere per centimetre.
Reservoir Rocks and Fluid properties
Permeability
Three types of permeability
Absolute permeability - the permeability of a porous medium with only one fluid present (single-phase flow).
When two or more fluids are present permeability of the rock to a flowing fluid is called effective permeability (ko, kg, kw).
Relative permeability is the ratio of absolute and effective permeabilities kro=ko/k, krg=kg/k, krw=kw/k.
A French hydrologist named Darcy did the first work on permeability. He was concerned about flow of water through filters. He found that flow rate Q, is proportional to area of flow A,h, and 1/L. This is expressed mathematically as flows:
1
Darcy’s “K” was determined to be a combination of◦ k, permeability of the sand pack (porous medium, e.g.
reservoir rock)◦ K is a constant of proportionality◦ , viscosity of the liquid
K constant may be written as;
2Reservoir Rocks and Fluid properties
Permeability
L
hhKAQ 21
μ
kK
Reservoir Rocks and Fluid properties
Alternatively, eqn.1; can also be expressed in terms of the pressure gradient dp over a section dL as;
3
Where dp = ∆hρg 4
– dp is the diff. b/w the upstream & downstream pressures ∆h the diff. b/w the upstream and downstream hydraulic
gradients ρ fluid density g is the acceleration due to gravity (9.81 m/sec2)
permeability
dL
dpKAQ
Darcy’sApparatus for DeterminingPermeability
Reservoir Rocks and Fluid properties
Permeability theory
outlet andinlet at head hydraulic &
medium porous theoflength
area sectional cross
rate flow c volumetri
21
21
hh
L
A
Q
AQL
hhQ
h1-h2
h1
h2
q
A
WATE
R
WATER
q
(Sand Pack Length) L
A
Reservoir Rocks and Fluid properties
Extension of Darcy’s Equation
• Darcy did not consider the fluid viscosity.
• Restricted to a medium with 100% saturated by water
• Other researchers discovered that Q is inversely proportional to
viscosity.
Assumptions Used in Darcy Equation:
1. Steady state flow, under laminar regime i.e. Qin = Qout
2. Viscous flow - rate of flow directly proportional to pressure
gradient
3. The flowing fluid is incompressible.
4. Porous media 100% saturated with fluid which flowed
5. Fluid and porous media not reacting
Reservoir Rocks and Fluid properties
Extension of Darcy’s Equation
Darcy’s Law can be extended to other fluids, if K is expressed as a ratio of /μ, where μ is the viscosity of a given fluid and the permeability of the porous medium.
Eqn. 3 can be written to account for viscosity μ as;
Q = - A 5
Eqn.5 can be integrated b/w the limits of length from 0 to L and pressure from P1(upstream) to P2 (downstream), for a fluid flow case.
Reservoir Rocks and Fluid properties
Extension of Darcy’s Equation
k
dL
dp
Therefore,
6
7
or 8
Hence, the original equation of Darcy was modified to account for viscosity as follows:
9 Reservoir Rocks and Fluid properties
Extension of Darcy’s Equation
ppdp
kdL
A
Q L2
10
ppkLA
Q12
0
21 ppL
KAQ
L
PKAQ
21 ppL
KAQ
Permeability is determined by:
Core analysis
Well test analysis (flow testing)
◦ RFT (repeat formation tester) provides small well tests
Production data
◦ production logging measures fluid flow into well
Log data
◦ MRI (magnetic resonance imaging) logs calibrated via core analysis
Reservoir Rocks and Fluid properties
Sources for Permeability Determination
• Shape and size of pore system
• Sorting
• Cementation
• Fracturing and solution
• Lithology or rock type
Reservoir Rocks and Fluid properties
Factors affecting the magnitude of permeability
Two sets of units are generally used in reservoir engineering: Darcy units and Oil-field units
For the purpose of the mathematical derivations, a system of units commonly referred to as Darcy units are used. These units are:
Vs = cm/sec; μ = cP; ρ = gm/cc K = Darcy; g = cm/sec2; dp/ds = atm/cm
For application to field data of the various mathematical expressions, the second system of units called the practical oil-field units are used.
Reservoir Rocks and Fluid properties
Units of permeability
From dimensional analysis, the dimension for K is [L]2.We could use ft2, ins2, or cm2 for measure of permeability, but these units are all too large to be applied in porous media.
So Darcy unit is used and recommended , but in many cases the millidarcy (mD)
1 Darcy = 1000 miliDarcy 1 Darcy ~ 10-8 sm2 (10-6 mm2) 1 miliDarcy ~ 10-11 sm2 (10-9 mm2)
Reservoir Rocks and Fluid properties
Units of Permeability
Darcy’s Law - Darcy Units
Linear (1-D) flow of an incompressible fluid
◦ where, q cm3/s k darcies A cm2
p atm cp L cm
◦ The Darcy a derived unit of permeability, defined to make this equation coherent (in Darcy units)
ΔpLμ
Akq
Darcy’s Law - Oilfield Units
Linear (1-D) flow of an incompressible fluid
◦ where, q bbl/D k millidarcies A ft2
p psia cp L ft
ΔpLμ
AkCq
Reservoir Rocks and Fluid properties
Common Oil Field UnitsQuantity Symbo
lDimension
Oilfield Units
SI Units
Mass m m Ibm Kg
Moles n n Ibmol Kmol
Force F ML/t2 Ibf N
Length L L ft m
Area A L2 acres m2
Volume-liquids v L3 bbl m3
Volume-gases v L3 ft3 m3
Pressure p m/Lt2 psi kPa
Temperature T T R K
Flow rate-liquids q L3/t bbl/d m3/d
Flow rate -gases q L3/t Ft3/d m3/d
Viscosity μ m/Lt cP mpa.s
Permeability k L2 md m2
Permeability is an important parameter that controls the reservoir performance. Its importance is reflected by the number of available techniques typically used to estimate it.
These different techniques provide formation permeability that represents different averaging volumes.
The quality of the reservoir, as it relates to permeability can be classified as follows
< 10 md fair10 – 100 High
100 – 1000 Very High
>1000 Exceptional
This scale changes with time, for example 30 years ago k< 50 was considered poor.
Reservoir Rocks and Fluid properties
Uses of Permeability
What is 1 Darcy?
A porous medium has a permeability of one Darcy, when a single-phase fluid of one centipoises viscosity, that completely fills the voids of the medium will flow through it, under conditions of viscous flow (also known as laminar flow), at a rate of one cubic centimeter cross sectional area, and under a pressure or equivalent hydraulic gradient of one atmosphere per centimeter.
Reservoir Rocks and Fluid properties
Definition of a Darcy
Important Characteristics that must be considered in the reservoir:
Types of fluids in the reservoir
Flow regimes
Reservoir geometry
Number of fluid flowing in the reservoir
Reservoir Rocks and Fluid properties
Permeability
Types of Fluids
• Incompressible fluids
• Slightly compressible fluids
• Compressible fluids
Flow Regimes
• Steady state flow
• Unsteady state flow
• Pseudosteady state flow
Reservoir Rocks and Fluid properties
Permeability
Reservoir Geometry
• Linear flow
• Radial flow
• spherical and hemispherical flow
Number of fluid flowing
• Single phase flow
• Two phase flow
• Three phase or multiple phase flow
Reservoir Rocks and Fluid properties
Permeability
Reservoir Rocks and Fluid properties
Types of Fluids
• Incompressible fluids
• Slightly compressible fluids
• Compressible fluids
Reservoir Rocks and Fluid properties
Flow Regimes• Steady state flow• Unsteady state flow • Pseudosteady state flow
Permeability
Reservoir Rocks and Fluid properties
Reservoir Geometry
• Linear flow• Radial flow• spherical and hemispherical flow
Steady state flow, i.e. Qin = Qout
Viscous or Laminar flow: the particles flow in parallel paths
Rock is 100% saturated with one fluid
Fluid does not react with the; this a problem with shaly-
sand (interstitial particles)
The formation is homogeneous and isotropic: same
porosity, same permeability and same fluid properties
Reservoir Rocks and Fluid properties
Assumptions Used in Darcy Equation
Generalized form of Darcy’s Law
The generalized Darcy’s law is given by the following equation:
Vs = Q/A and dp/ds = dp/dx, hence
Reservoir Rocks and Fluid properties
Horizontal, Linear, Liquid System
ds
dpk
Vs
dx
dpKAQ
Separating the variables and integrating gives the following eqn.:
Note that P1>P2
2112
0
6
0
10*0135.1
2
1
PPk
PPk
LA
Q
dPk
dxA
Q
dPk
dxA
Q
dx
dPk
A
Q
dx
dPkv
ds
dzg
ds
dPkv
A
Q
P
P
L
x
s
• Fluid not compressed (no density
change with pressure)
• Steady flow (mass in = mass out)
• In horizontal flow (Linear):
dx
dP
ds
dP
ds
dz , 0
L
PPkAQ 21
Reservoir Rocks and Fluid properties
Flow Potential
ds
dz
ds
dpk
A
qv
c
gss
The generalized form of Darcy’s Law includes pressure and gravity terms to account for horizontal or non-horizontal flow
The gravity term has dimension of pressure / lengthFlow potential includes both pressure and gravity terms, simplifying Darcy’s Law
ds
dΦ
μ
k
A
qvs
= p - gZ/c; Z+; Z is elevation measured from a datum has dimension of pressure
The gravitational conversion constant, c, is needed to convert the gravity term to units of pressure gradient [pressure/length]
A clear way to determine the sign of the gravity term (+ or -) is to consider the static case. If no fluid is flowing, then there is no potential gradient (potential is constant), while pressure changes with elevation, due to fluid density (dp=(g/c)dZ).
Reservoir Rocks and Fluid properties
Flow Potential
Reservoir Rocks and Fluid properties
Reservoir Rocks and Fluid properties
Fluid not compressed Steady flow (stream mass was constant)
Radial Flow System
rw
re
Pwf Pe
h
r
w
e
we
wew
e
PP
rr
P
P
r
r
rrPPhk
Q
PPk
r
r
h
Q
Pk
rh
Q
PdPk
drrh
Q
e
w
e
w
e
w
e
w
ln
2
ln2
ln2
1
2
Q = flow rate, m3/sec K = permeability, m2
h = thickness, m Pe = pressure drainage radius, N/m2
Pwf = flowing pressure, N/m2
= fluid viscosity, N/m2
re = drainage radius, m rw = the well-bore radius, m
Reservoir Rocks and Fluid properties
Horizontal, Radial, and Liquid System
Linear, Parallel Flow
◦ Discrete changes in permeability
◦ Same pressure drop for each layer
◦ Total flow rate is summation of flow rate for all layers
◦ Average permeability results in correct total flow rate
i321 qqqqq
32121 ΔpΔpΔpΔpp-p
i321 hhhhh
• Permeability varies across several horizontal layers (k1,k2,k3)
hwA;ΔpLμ
hwkq
• Substituting,
• Rearranging,
• Average permeability reflects flow capacity of all layersh
hkk ii
ΔpLμ
hwkΔp
Lμ
hwkΔp
Lμ
hwkΔp
Lμ
hwkq 332211
Serial Flow
◦ Discrete changes in permeability
◦ Same flow rate passes through each layer
◦ Total pressure drop is summation of pressure drop across layers
◦ Average permeability results in correct total pressure drop
321 qqqq
i32121 ΔpΔpΔpΔppp
i321 LLLLL
hwA;hwk
Lμqp-p 21
• Permeability varies across several vertical layers (k1,k2,k3)
Linear, Serial Flow
hwk
Lμq
hwk
Lμq
hwk
Lμq
hwk
Lμqp-p
3
3
2
2
1
121
• Substituting,
• Rearranging,
• If k1>k2>k3, then– Linear pressure profile in each layer
i
i
kL
Lk
x 0 L
p
p1
p2
0
k
Radial, Parallel Flow
◦ Discrete changes in permeability
◦ Same pressure drop for each layer
◦ Total flow rate is summation of flow rate for all layers
◦ Average permeability results in correct total flow rate
i321 qqqqq
321we ΔpΔpΔpΔpp-p
i321 hhhhh
• Permeability varies across several (3) horizontal layers (k1,k2,k3)
Δp)/rln(rμ
hk2πq
we
Radial, Serial Flow• Substituting (rw=r1, r2 ,re=r3),
• Rearranging,
LayersAll i
i1i
we
k)/r(ln(r
)/rln(rk
hk2π
)/rln(rμq
hk2π
)/rln(rμq
hk2π
)/rln(rμqp-p
2
2e
1
w2wewe
Reservoir Rocks and Fluid properties
Permeability
It is necessary to determine an average value of permeability. Three common types of computed averages are as follows:
i) arithmetic average
ii) harmonic average
ii) geometric average
Selection of the averaging technique should be based primarily on the geometry of the flow system.
Reservoir Rocks and Fluid properties
Averaging Permeability
k1
k2
k3
h1
h2
h3
i
iiA h
hkk
Q
Arithmetic Average
Parallel Flow
Reservoir Rocks and Fluid properties
Averaging Permeability
L1 L2 L3
k1 k2 k3
ii
iH kL
Lk
/
Harmonic Average
Series Flow
Reservoir Rocks and Fluid properties
Averaging Permeability
ihhhh
G kkkk1
321 .......321
Geometric Average
Random Flow
Reservoir Rocks and Fluid properties
Interaction between porosity & permeability
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