1Supplement 2More on Permeability

What is Permeability?

Absolute permeability is a measure of the capacity of a particular rock to flow (circulate or transmit) a single fluid contained in the pores (under the action of an applied pressure gradient). Defined by Darcys Law (horizontal flow)

dldpkAq =

PERMEABILITY Permeability is a property of the porous medium and a measure

of the mediums capacity to transmit fluids.

A porous medium has a permeability of one darcy (D) when a fluid of one centipoise viscosity will flow at a rate of one cubic centimeter per second per square centimeter of cross-sectional area when subjected to a pressure gradient of one atmosphere per centimeter.1 darcy = 1000 millidarcy (md) = 9.869x10-13 m2

* The permeabilities of the formations vary from about 0.1 to 1000 md or higher.

qA

q

pL

Porous medium

pALqk =

2PERMEABILITY

The permeability is a function of the degree of interconnection between the pores in the rock. This figure illustrates two samples having the same porosity. Evidently fluid will flow from north to south much more readily through sample A than through sample B, where the flow is restricted by the fine capillaries.

Note: Capillaries are flow channels or tubes with very fine diameters (~0.001-0.1 cm)

A B

BA

BAkk >

=

Tilted Axis

+Z'g

gm/cc

+Z(L1)

+Z(L2)

+L

L1

L2

lhKq =

p1

p2

Tilted Axis

Suppose that the direction of flow is tilted to the vertical.

Let z denote the direction in which gravity acts , then velocity of fluid is

( ) ( )

=12

121212

LLLzLzg

LLppkv

( ) ( )

=12

121212

LLLzLzg

LLppkAq

3Sign Convention

If q > 0, it means fluid is flowing in the positive L direction, i.e., downward. If q < 0, it means fluid is flowing in the negative L direction, i.e., upward.

= 12

12

12

12

LLZZg

LLppkAq

Differential form

Eqs. on preceding slide assume steady-state flow and the k, A, viscosity and density are constant. Applies for any liquid, not just water.

Letting L = l =L2 L1 0

= dlzdg

dldpkv

= dl

zdgdldpkAq

Darcy Equation in Field Units

Eqs. on preceding slide assume consistent system of units (e.g. SI or cgs). In oilfield units, Darcys equation is given by

DftRB

dlzd

gg

dldpkxv

c

= 23 14410127.1

DRB

dlzd

gg

dldpkAxq

c

=

14410127.1 3

4Darcy Equation in Field Units We define to be specific weight (or pressure

gradient) of a fluid:

Then Darcys equation can be expressed in terms of specific weight of the fluid as

ftpsi

gg

c

=144

DRB

dlzd

dldpkAxq

=

310127.1

In Terms of Angle If is the angle of bedding plane with

the horizontal measured from the horizontal to the l axis in the counter clockwise direction, then

= sindlzd

+l-direction

+z-axis

Horizontal

In Terms of Angle

If is the angle of the bedding plane with the horizontal measured from the horizontal to the l axis in the counter clockwise direction, then

= dlzd

dldpkAxq

310127.1

+= )sin(10127.1 3 dldpkAxq

5Note on the Angle

Always remember that is the positive angle measured from the horizontal to the positive l axis in the counterclockwise direction.

If you measure it in the clockwise direction, then is a negative angle.

Surface vs In-situ Rates As we usually measure flow rates at surface, when

modeling flow in a reservoir, we should relate surfacerates to rates in the reservoir.

We use formation volume factors (FVFs). For examplefor an oil reservoir producing above bubble point, if theoil flow rate at stock tank conditions is qosc in STB/D and the FVF is Bo, then the flow rate of oil in reservoirconditions, qo, is

DRB

STBRB

DSTBBqq oosco ==

DSTB

RBSTB

DRB

Bqq

o

oosc ==

Surface vs In-situ Rates

Then for a linear-horizontal flow in an oil reservoir above the bubble point, then the surface flow rate can be expressed by using Darcys equation as

whA == and0

qo

P(0)

w

h

k

0 L

P(L)

( ) ( ) ( )

+=

c

o

ooosc g

gL

pLpBkAq

144sin010127.1 3

6Multidirectional Flow

In Darcys experiment, we have assumed that the flow is only in the L direction. This means that for a fixed value of L,

pressure was constant throughout the cross-sectional area of the sandpack.

When you eventually consider reservoir flow, we often need to consider the three components of velocity

Permeability depends on direction

Permeability Grain Size

The shape and size of the sand grains are important in determining the size of openings between the sand grains Elongated grains higher permeability in

direction of elongation Rounded grains large permeability in all

directions In general, smaller grain size implies smaller

permeability

7Permeability Grain Size

Permeability vs. Porosity

Generally, we cannot say that permeabilityand porosity are related.

However, in sandstone reservoirs, typically as porosity increases, permeability increases.

When carbonates are considered, we maynot find any relation between permeabilityand porosity.

Also, shale is very porous, but no permeability.

Permeability vs. Porosity

8Poiseuilles Law

Capillary Flow

Model of Porous Medium

Pore spaces treated as if they were made up of bundles of capillary tubes of various diameters Only an approximation to true physics Helps us to understand flow in porous media

Capillary FlowL

2r0

Laminar (Low velocity) Flow Concentric rings of fluid flow at the same rate Ring closest to the walls of capillary have zero

velocity Each ring of fluid drags against neighboring

rings Ring at the center has maximum velocity

9Viscous (Shear) Force

Shear force for ring of length L

Viscosity, poise Length, L cm Velocity gradient cm/s/cm Force - dyne

drdvstressshear

drdvrLF == )(2

Viscous Force

Shear force for a fluid element of radius rcm and length L cm will be:

Must have a pressure drop across length of tube to overcome shear force. Can show

( )drdvrLF 2=

( ) 02 2 =+ prdrdvrL

Velocity Profile

Can solve for velocity profile by integrating and using the fact that v = 0 at the capillary wall. Integrate

from

to obtain

( ) ( ) ( )Lprrrv = 4

220

0rtor( )Lpr

drdv

2=

10

Velocity Profile

Volumetic rate of flow through a ring of thickness dr is

Integrate the last expression to obtain total rate

( ) ( ) ( )L

pprrrv 2122

04

=

rdrvvdAdq 2==

Volumetric Rate Total Rate obtained by integrating over all radii

This is the Hagen-Poiseuille equation for viscous flow through a capillary tube, in any absolute units. Also applies in Darcy units.

( ) ( ) = 00

220 2

4

rrdr

Lprrq

( )Lprq =

8

40

Example

Find the flow rate of a 0.015 poise fluid through a capillary 0.10 cm in radius and 25 cm long under a differential pressure of 300 dyne/cm2

( ) sec/cm 0314.025

300015.081.0 34 ==

q

11

Similarity to Darcys Law

Darcys law for linear flow and Hagen- Poiseuillelaw are quite similar.

Darcys Law in terms of cc/sec, poise, cm, dyne/sq cm and darcies (instead of sq cm) becomes

Hagen-Poiseuille law

( )L

pprkq 212

091086.9

=

( )L

pprq 214

0

8

=

Equating Laws

Equivalent permeability

Radius here is in cm In terms of diameter in inches

darcies . 20610712 rk =

darcies 2Dk 61020 =

Bundles of Capillaries

If the rock is assumed to consist of n capillaries bundled together, then total rate is given by

is the radius of the ith capillary tube.

( ) ==n

i

iLprq

1

4

8

ir

12

Bundles of Capillaries

For cross-sectional flow area A cm2, equivalent permeability is found by equating flow rate from Poiseuilles law and Darcys Law:

( )=

=n

i

i

Lpprq

1

214

8

darcies in . ==n

iirA

k1

4610712

( )L

ppAkq 21910869 =

.

Bundles of Capillaries

If all terms including k are in cgs units, the equations from the previous slide become:

( ) ==n

i

iLprq

1

4

8 ( )

LpkAq =

2cm ==n

iirA

k1

48

Note Thus, we can show that for cubic packing of uniform

spherical grains, the radius of a pore throat (equivalent to a capillary), is directly proportional to grain radius. Thus, Poiseuilles law suggests that absolute permeability

depends on the size of the pore throats, or equivalently, on thesize of the grains.

Porosity of this type of system is independent of grain size only depends on packing

Porosity and Permeability are not directly related. Nevertheless, engineers often try to correlate porosity and permeability by plotting ln(k) versus . This plot often appears to show that a correlation exists,

particularly for sands.

13

Permeability Terms

It is important to know what the terms homogeneous, heterogeneous, isotropicand anisotropic mean.

We say that the permeability field is isotropicif kx = ky = kz.

If permeability depends on direction, we say permeability is anisotropic.

Occasionally you hear the term, areallyisotropic which means kx = ky, but kz may be different.

Permeability Terms

Areally isotropic In such cases, we often simply use k to

denote horizontal permeability and kz to denote vertical permeability. Having kz = 0.1kis not uncommon.

The terms isotropic and anisotropicapply only to parameters or functions that may depend on direction.

Permeability Terms

The term homogeneous can be applied to either porosity or a component of permeability

That porosity is homogeneous in a region of the reservoir means the porosity is uniform, i.e., does not depend on position.

If we say the permeability field for a reservoir is homogeneous and isotropic, we mean kx = ky= kz = k and k does not depend on position (location).

14

Permeability Terms

A permeability (kx , ky , or kz) is said to be heterogeneous if it varies with position.

The porosity field is said to be heterogeneous if varies with position, i.e., is not homogeneous.

In general, we simply say permeability is heterogeneous which means all three permeabilities may vary with position, or homogeneous which simply means all permeabilities are independent of location.

Homogeneous and Isotropic

Isotropy:A: kax=kay , kax=kaz , kay=kazB: kbx=kby , kbx=kbz , kby=kbz Homogeneity:

A Bkax=kbx , kay=kby , kaz=kbzConclusion:

kax=kay = kaz=kbx = kby=kbz

A

B

kaxkay

kazkbx

kby

kbz

Homogeneous and Anisotropic Anisotropic:A: kaxkay , kaxkaz , kaykazB: kbxkby , kbxkbz , kbykbz Homogeneity:

A Bkax=kbx , kay=kby , kaz=kbzConclusion:

kax=kbx kaykaz , or kbykbzkay=kby kaxkaz , or kbxkbzkaz=kbz kaxkay , or kbxkby

A

B

kaxkay

kaz kbxkby

kbz

15

Heterogeneous ve Isotropic Isotropic:

A: kax=kay=kaz

B: kbx=kby =kbz Heterogeneous:

A B

kaxkbx , kay kby , kaz kbz

Conclusion:

kax=kay=kaz kbx or kbyor kbz

kbx=kby=kbzkax or kay or kaz

A

B

kaxkay

kaz kbxkby

kbz

Heterogeneous and Anisotropic Anisotropy:A: kax kay kaz B: kbx kby kbz Heterogeneity:

A B

kax kbx , kay kby , kaz kbzConclusion:

kax kay kaz kbx kby kbz

A

B

kaxkay

kaz

kbxkby

kbz

Note

Heterogeneities in permeability can have a dominant effect on secondary recovery operations. Thin high-permeability sand streaks can

cause water to bypass oil sand and break through prematurely at production wells during waterflooding.

16

HeterogeneityPermeability field (cross-section) generated from a statistical model

Upscaled 3D model