Permeability

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


Permeability theory

Permeability is an INTENSIVE property of a porous medium (e.g. reservoir rock)

Permeability


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.


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


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


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


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



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.



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


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


Sources for Permeability Determination

• Shape and size of pore system

• Sorting

• Cementation

• Fracturing and solution

• Lithology or rock type


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.


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)


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


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.


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.


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


Permeability

Types of Fluids

• Incompressible fluids

• Slightly compressible fluids

• Compressible fluids

Flow Regimes

• Steady state flow

• Unsteady state flow

• Pseudosteady state flow


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


Permeability


Types of Fluids

• Incompressible fluids

• Slightly compressible fluids

• Compressible fluids


Flow Regimes• Steady state flow• Unsteady state flow • Pseudosteady state flow

Permeability


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


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


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


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).


Flow Potential



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


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


◦ 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


◦ 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


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.


Averaging Permeability

k1

k2

k3

h1

h2

h3

i

iiA h

hkk

Q

Arithmetic Average

Parallel Flow



L1 L2 L3

k1 k2 k3

ii

iH kL

Lk

/

Harmonic Average

Series Flow



ihhhh

G kkkk1

321 .......321

Geometric Average

Random Flow


Interaction between porosity & permeability

Permeability

Engineering

fluid viscosity

fluid propertieslength

aflowing fluid

fluid flowcase

given fluid

fluid propertiesviscosity

fluid propertiesdl

fluid propertieskk