1Absolute Permeability

What is it?

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.

dldpkAq =

Darcys Law

For flow through a horizontal sand pack, flow rate is Directly proportional to the pressure drop across

the pack Directly proportional to the (gross) area open to

flow Inversely proportional to the length of the pack Inversely proportional to fluid viscosity

Constant of proportion is the permeability

Darcys Experiment, 1856

Darcy was a civil engineer who was designing a water filtration system for the city of Dijon.

Needed to determine the flow rates possible through sand-pack filters for a given applied pressure drop.

First experimental Reservoir Engineer.

2DarcysApparatus

SandPack

L1

L2

L

d1

d2

p1

p2

pa

paConstant ratewater injection

hf = h1

h2

Experiment

Water flowed at a constant rate qcm3/sec (in L direction) through a sand pack with area A cm2. Water velocity, v = q/A cm/sec

Manometers (water) placed at top and bottom of sand pack

When flow stabilized, he found

Lhh

Lhv f

= 2

Darcys Result

K is independent of the length of the sand pack and the velocity Depends only on the sand in the sand pack

Heights and L are in units of centimeter; pressure is in dynes/cm2, density in g/cm3 and g = 980 cm/sec2

=L

hhKv f 2

Darcys Law in terms of Pressure

From basic Physics results

Geometry of apparatus

11 gdpp wa +=SandPack

L1

L2

L

d1

d2

p1

p2

pa

paConstant ratewater injection

hf = h1

h2

22 gdpp wa +=( )2121 ddgpp w =

221 dhLdhf ++=

3Combining Equations

gppdd

w= 2121

221 dhLdhf ++=

( )Lgppg

Lgpphh

ww

wf

+=

+=

21

212

1

+

=

= g

Lpp

gK

Lhh

Kv ww

f 212

Modifications to Original Work Darcys experiments were conducted with

water only For fluids of different density and viscosity

k has cm2 as its units, or more generally the same units as area

+= gLppkv

21

121212 LLLg

LLppkv =

=

UnitsParameter Symbol Dimensions cgs SI Darcy Field

Length L L cm metre cm ft

Mass m M gm kg gm lb

Time t T sec Sec sec hr

Velocity u L/T cm/sec metre/sec cm/sec ft/sec

stb/d

(liquid)

Rate q L3 /T cc/sec metre3 /sec cc/Sec

Mscf/d

(gas)

Pressure p (ML/T2 )/L2 dyne/cm2 Newton/metre2

(Pascal)

atm psia

Density M/L3 gm/cc kg/metre3 gm/cc lb/cu.ft Viscosity M/LT gm/cm.sec

(Poise)

kg/metre.sec cp cp

Permeability k L2 cm2 metre2 Darcy mD

In Terms of Rate

Suppose rate was zero:

= gLLppkAq 12

12

01212 =

gLLpp

Lgpp += 12

4Sign 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.

= gLLppkAq 12

12

Tilted Axis

Tilted Axis

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

Let z denote the direction in which gravity acts ( ) ( )

=12

121212

LLLzLzg

LLppkv

( ) ( )

=12

121212

LLLzLzg

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

=

lzg

lpkv

=lzg

lpkAq

5In Terms of Angle of Inclination of Bedding Plane

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

=

lzg

lpkv

=

lzg

lpkAq

+

= )sin(glpkv

+

= )sin(glpkAq

Multidirectional Flow

In Darcys experiment, we have assumed that the flow is only in the Ldirection. 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

x - y - z coordinate system, SI or CGS units

The three components of velocity are

=

xzg

xpkv xx

=

yzg

ypkv yy

=

zzg

zpkv zz

x - y - z coordinate system , field units, velocity RB/sq. ft-day The three components of velocity are

= xz

gg

xpkv

c

xx 144

10127.1 3

= yz

gg

ypkv

c

yy 144

10127.1 3

= zz

gg

zpkv

c

zz 144

10127.1 3

6Permeability and Direction

Permeability can depend on direction In fact, if the rock under consideration was

originally deposited from a flowing river (assumed to be straight here) and we choose the x coordinate to follow the path of the river, it is not unusual to find that kx > ky. In most cases, vertical permeability (kz) is lower that horizontal permeability, usually because streaks of shale exist which retard vertical flow.

Permeability and 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

Permeability Grain Size

7Permeability 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 = 0.1k is 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).

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.

8Scale Whether porosity may be considered

homogeneous or not depends on the scalethat will be used to make reservoir calculations. If we cut a small core sample at every foot along a

vertical well in an oil reservoir, typically there would be considerable variation in porosity (and horizontal and vertical permeability) from core to core, but on a scale of 100 feet by 100 feet by 10 feet that might be used to make calculations using a reservoir simulator, the porosity (or permeability) variation might be so small that we might consider permeability homogeneous.

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.

Cylindrical Coordinates

In the cylindrical coordinate system, it is not convenient to consider a permeability in the direction so we will assume that permeability is areallyisotropic (k)

rpkvr

=

=

prkv

= gz

pkv zz

Global Flow Regimes

At any given time in the producing life of a reservoir, the fluid flow condition existing may be characterized as either a) transient, b) pseudosteady state (sometimes, but not by me, called semisteady state) I call it boundary dominated flow or c) steady state.

What do these terms mean?

9Reservoir Flow

Initially, in a virgin reservoir, the pressure at any fixed depth is constant.

As production begins, the pressure near the wellbore drops significantly as near-wellbore fluids expand to satisfy the imposed production condition.

Far away from the well, no measurable pressure drop can be observed at early times locations far away from the well are not aware

that the reservoir is being produced.

Reservoir Flow As time progresses, pressure drops can be

measured further and further away from the well, an increasing volume of the reservoir fluids expand to contribute to the well's production.

During this period, the reservoir is said to be infinite acting and the flow is transient; pressure drop at outer reservoir boundary is negligible.

The pressure versus time behavior at the producing wellbore contains information about the reservoir permeability.

Reservoir Flow After a long time, pressure drops can be

measured at all reservoir locations the entire reservoir is contributing to the well's

production. At this time, the pressure changes at the

same rate at every location in the reservoir, i.e., dp(x,t)/dt = constant; pseudosteadystate or boundary dominated flow.

The pressure versus time behavior at the wellbore reflects the volume of fluid (or the reservoir pore volume) contributing to production.

Reservoir Flow

To see steady-state flow in a reservoir, we must replace reservoir fluids at the same rate that we remove them.

This situation occurs in secondary and enhanced oil recovery operations - e.g., waterflooding, gas injection, etc.

During steady-state single-phase flow, nothing is changing in the reservoir, i.e., dp(x,t)/dt = 0, so pressure versus time data contains no reservoir information.

10

Examples

Darcys Law

Steady-State Flow

For steady-state flow in a linear bed, Darcys Law is given by

( ) ( ) ( )

+=

csc g

gL

pLpBkAq

144sin010127.1 3

( ) ( ) ( )

+==

c

sc

gg

LpLpk

ABqv

144sin010127.1 3

22 232232

sec.sec/. ft

lblbgftg

f

massc ==

Example 1

Assume flow of fresh water from left end (l = 0) to the right end (l = L =4) feet through a sand pack in a horizontal tube of diameter 4 in. Assume formation volume factor is 1.0.

Assume steady-state flow with the pressure at the left end 40 psi, the pressure at the right end 15 psi and the flow rate 1.5 barrels/day. Assume the viscosity of water is 1 cp. Estimate the permeability in millidarcies.

Solution

The density of fresh water is 62.4 lb-mass/ft3 (1 gm/cm3) so the specific weight of water is

The cross-sectional area of the sandpack is

psi/ft. 4333.02.321442.324.62

144=

==c

ww g

g

.ft 08727.0122 2

2

=

=A

11

----------------

From Darcys Law

Solving for permeability gives k = 2,440 md.

( ) ( )

+

= 0sin4333.04

40151

08727.010127.15.1 3 k

( ) ( ) ( )

+=

csc g

gL

pLpBkAq

144sin010127.1 3

Example 2

Now assume the tube is inclined at an angle /6. Estimate the flow rate.

From Darcys Law ( ) ( )

bbl/day .

sin...

4471

30433304

401510872702440101271 3

=

+

= q

Steady-State Flow

For steady-state linear flow rate

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.

( ) ( ) ( )

+=

csc g

gL

pLpBkAq

144sin010127.1 3

Important Note

is the angle measured from the horizontal to the positive l axis in the counterclockwise direction

The negative sign in the preceding flow equations is such that velocity and flow rate are positive if flow is in the positive l direction, whereas velocity and flow rate are negative if flow is in the negative l direction. You are free to choose the direction of l.

12

Example 2

Now assume the tube is inclined at an angle /6 downward . Estimate the flow rate. Then from Darcys Law,

( ) ( )bbl/day .

sin...

551

30433304

401510872702440101271 3

=

+

( ) ( ) =

+

= 330433304

401510872702440101271 3 sin...q

Steady-State Linear Flow

Remember, you are free to choose the l-axis, but p(0) always refers to the pressure at the origin of the l-axis. If the axis goes from left to right though the core or sandpack (positive direction is to the right), then p(0) is at the left end of the core and p(L) is at the right end. The choice of the l-axis affects .

( ) ( ) ( )

+=

csc g

gL

pLpBkAq

144sin010127.1 3

Steady-State Linear Flow

If the axis goes from right to left, i.e., positive direction is to the left, then p(0) is at the right end of the sandpack and p(L) is at the left end. The choice of the l-axis affects the angle which is always measured from the horizontal to the positive l-axis in the counterclockwise direction.

( ) ( ) ( )

+=

csc g

gL

pLpBkAq

144sin010127.1 3