Lecture 4 (Ch5) FunRE 2009

8/2/2019 Lecture 4 (Ch5) FunRE 2009

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Reservoir EngineeringFundamentals

Semester 1 2009

Permeability

Permeability and Porosity

Gas and Liquid Permeability

Anisotropy

Relative Permeability

Lecture 5: Darcys Law and Applications


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Permeability

Darcys Law quantifies the rate of flow of fluidsthrough porous media via the concept of permeability.

For one-dimensional, horizontal flow through aporous medium Darcys Law states that:

q = rate of fluid flow (cm3/s)A = cross-sectional area open to flow (cm2)

= viscosity of flowing fluid (cp)

p = pressure drop across porous medium (atm)

L = length of porous medium (cm)k = permeability (Darcy)

pL

Akq

= A

L

qq

1p

2p

p

k is a constant defined as permeability

Permeability

One Darcy permeability will yield a flow ofapproximately one barrel/day of one centipoise oilthrough one foot of formation thickness in a well borewhen the pressure differential is one psi.

The relationship between flow rate andpressure change can be adapted for anytype of flow geometry.

er

wr

ep

wp

erwr

h)/ln(2.141

)(

we

we

rr

ppkhq

=

For example, a radial flow geometry


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Assumptions in Darcys Law

1. Flow is laminar i.e. fixed flow paths (streamlines).

3. One phase present at 100% pore space saturation.

2. No (chemical) reaction between fluid and rock.

Assumption 1: Laminar Flow

Generally reasonable assumption in reservoir;

typical flow rates too low to promote turbulent flow.

Turbulent flow can occur around well bores of high

rate wells, results in deviation from Darcys Law.


This deviation from Darcys Law is referred toas Non-Darcy Flow.

Forchheimer introduced a velocitysquared term to describe the Non-Darcy behaviour at high pressure

differentials.

rate

Non-Darcy Flow

Darcy Flowp

2vv

kdL

dp

+=

= inertial co-efficient tubulence factor -dependent on fluid and rock properties

Darcy Forchheimer


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Assumption 2: Chemical Reactions between rock & fluid

If there are changes to the structure of the porousmaterial eg. clay swelling, or dissolution of the rock,Darcys law will be inadequate to describe fluid flow.

Two and three-phase flow are common in reservoirs.

Assumption 3: Single phase flow

Darcys law can be extended to multiphase flow usingthe concept of relative permeability.

Permeability and Porosity Relationship

Environmental & depositional factors influencing porosity alsoeffect permeability, and often there is a relationship.

Relationship is not simple, porosity is avolume, permeability controls flow offluids.

Permeability Depends on: ermea

ty

(md)

Porosity (%)

0 5 10 15 20 25 30 35

10

1

100

1000

ReefLim

esto

ne

Oolithic

Lim

esto

ne

Suc

rosicD

olom

oite

Cha

lkyLi

mesto

ne

Fin

eG

rain

ed

Friable

Sand

We

llC

emented

HardS

and

Inte

rcry

stallin

e

Lim

esto

nean

d

Dolomite

Typically, increased permeability is

accompanied by increased porosity.

BUT

1. Internal surface area of the pore space.

2. Distribution of pore throat sizes.

3. Tortuosity of fluid flow paths.


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Kozeny (1927) derived an equation to improve therelationship between porosity & permeability.

From Hagen-Poiseuilles law:

The first assumption was to consider a porous rock toconsist of a bundle of capillary tubes of equal length.

L

rq

p

L

prq

=

8

4

Comparing the above equation with Darcys law it can be seen that the

effective permeability of a single horizontal tube is:

82rk=pL

Ak

L

pr=

8

4


The porosity () of a bundle of (n) capillaries whose endsoccupy a surface area of (A) is given by:

From the Hagen-Poiseuilles equationthe flow rate forn capillaries is:

Arn2= 2rAn =&

L

prnq

=

8

4

Substitutingn & comparing with Darcys Law gives:

8

2rk

=

The permeability of reservoir rock will depend on porosity andthe square of the pore throat size.


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This approach can be modified to allow for tortuosity, , of thereal pore network, such that the actual length of each capillary

is L, to give:

Permeability is impacted by surface area because of viscousdrag, & is related by the specific surface.

The specific surface, , is defined as the interstitial surface area of the

pores per unit of bulk volume, & for the example ofn capillaries is:

8

2r

k=

Arn 2=

r2=

substituting forn from previously:2rAn =


Substituting for r from this equation into the equation relating

permeability, porosity,r & yields the Kozeny equation:

In a more general form this equation becomes:

Whereck is the Kozeny constant

2

3

2 =

k

2

3

=

kck

r2=


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Kozeny equation incorporates parameters which control fluidflow (tortuosity & pore surface areas) to improve therelationship with porosity.

a

w

For the example of a simple cubic structure

( )2

32

1

=

rck scsc

For the example of a simplified fracture model

a

wc

a

a

w

ck fff

3

2

3

1=

=

Anisotropy- Directional Permeability

Permeability is a directional quantity and increases difficulty indescribing the permeability distributions of naturally porousmaterials.

Permeability anisotropy is related to depositional controls.

Permeability is lower in the

plane that is normal to thedepositional plane because ofgrain alignment.

x

zy

Currentdirection (y)

Direction of maximum

permeability (ky)

Permeability inxdirection (k

x) < k

y

Vertical permeability

(kz) <


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Gas flow in a porous medium differs from liquid flow &permeability measured in a gas flood experiment will be differentto that of a liquid flood.

Many laboratory experiments are conducted using gas so it isnecessary to convert to a liquid system if this is the situation inthe field.

liquid

liquid molecules in thecentre of the pores move ata higher velocity

As an ideal viscous liquid (laminar flow) flows over a solidsurface it adheres to the surface such that the fluid velocity at

the surface is zero.


However, this does not apply for gases since gas does not stickto the walls.

Taking the experimental set-up for an arbitraryfluid (as shown) & from the ideal gas law forisothermal conditions it follows that:

aP

bP

cross sectional area,A

flow rate, qa

L porous medium

flow rate, qb

( )[ ]gZppLkA

q ba +

=

aa

ba pqpp

q =

+

2

Eventually, the modified Darcy equation for gas becomes:

To help illustrate this take Darcys Law & include gravity(to account for vertical flow):

=

L

ppkApq ba

aa2

22

Mean pressure


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The difference in molecular movement as compared with liquidflow results in a dependence of gas permeability on the meanpressure of the gas existing at the time of measurement.

A flowing gas does not adhere to the surface of the porousmedium, which is a requirement of Darcys law, and aphenomenon termed slip occurs.

gas

gas molecules flow at a

uniform rate through smallpores


Therefore permeability measured from a gas flood will begreater than when measured using a liquid.

In contrast, the permeability of a rock to a given liquid is aconstant and is independent of the pressure differentialimposed.

The dependence of gas permeability on pressure was notedby Klinkenberg and is usually referred to as theKlinkenberg effect.

+=

p

bkk lg 1 At lowp, b > 0

p

increases slipand differencebetweenkg &kl

0slip isnegligible

At highp, bp

b=const. Charcteristic of gas & the porous medium)


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Values converge at high pressures, & relative correction ishigher for low permeability samples .

0.20

10

20

30

40

50

0

Reciprocal Mean Pressure,pm: (Atm)-1

GasPermeability:(md)

lk

0.4 0.6 0.8 1.0

( )mlg pbkk += 1

Klinkenberg'sstudies alsorevealed thatgas

permeabilityis a functionof the gascomposition.

1 2 3 4 50

20

40

60

80

100

0

Hydrog

en

Nitrog

en

Carbon

Dioxide

Reciprocal Mean Pressure,pm:(Atm)-1

GasPermeability:(md)

lk

( )mlg

pbkk += 1

Rock-Fluid Reaction

Reaction between reservoir rock and the fluids introduced canincrease or decrease permeability.

20000

20

40

60

80

100

0

Clean Sand

Water Salinity: Parts per Million

WaterPermeability:%ofAirPermeability

4000 6000 8000 10000

Moderate ClayContent

High ClayContent

100

10

100

0

50,000ppm

Air Permeability: md

WaterPermeability:md

100 1000

Fesh Water

1000

For example, anhydrate dissolution increases permeabilitywhereas clay swelling, as a result of the introduction of water,particularly fresh water, reduces permeability.


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Rock-Fluid Reaction

It can be the case that particle movement rather than clay swellingis the dominant mechanism reducing permeability. Although ifclays are present it is generally a combination of both.

100

20

40

60

80

100

0

ParticleMovement

Brine Injected: Pore Volumes

BrinePe

rmeability:%ofOriginal

20 30 40 50

Flow DirectionReversed

Clay Swellingand / or

Particle Movement

Permeabilityreduction due toa combination

of clay swellingand particlemovement

Particles dislodgedwith reverse flow,now just seeing

damage effects ofswelling

Permeability

now furtherdecreased due

to particlemovementReverse Flow


Absolute permeability (k) is the capacity of a rock to transmitfluid under a pressure differential when its pore space iscompletely saturated with the fluid.

Darcys law can be extended to accommodate multiphase flowby measuring the effective permeability of each phase.

Relative permeability is a normalised measure of conductanceof one phase in a multiphase system. It is a measure of themutual interference between phases competing for the samepore space.


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Sw

Swc

1-Sor

ko kw

10

k kabsolute permeability

whenSw=S

wc

kw= 0

whenSw=1

kw= k

whenSw=0ko

= k

@Sor

ko

= 0

(kw,ko)


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Sshaped curve indicates oil is the non-

wetting phase because the reduction inkro atlow Swis not significant.

Fluid and rock properties influence the quantity of residual saturationsand the shape of the relative permeability curves.

Sw

Swc

1-Sor

kro krw

1 1

10

k'rw

k'ro

Even for relatively modest changes in water

saturation, there will be large changes inwater relative permeability. Non-wettingphase will sit in the largest pore spaces

effectively blocking the flow of the wettingphase

krw< 1, because the surface area for drag

seen by the water is higher than oil. Actualvalue dependant on wettability.


Each fluid establishes its own set of tortuousflow paths and a unique set of channels appearto be established for each relative saturation

WaterOil

smallrc

highPc

Water

Oil

greaterrclowerP

c

Water

Oil

largerc

such thatPc

too low to maintain

continuitythrough pore throat,"snap-off" occurs

As the saturation of the non-wetting phase (oil) isreduced the channels forthis fluid tend to break down

Eventual formation of isolated islands of non-wetting fluid which cannot be removed usingreasonable pressure gradients, corresponding to

laminar flow. This is the origin of residual oil


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Wettability is reflected in the shapes of relative permeability curves.

0.5

Sw

0.5

k'r

1.0

1.0

Oil

Water

Water - wet

0.5

Sw

0.5

k'r

1.0

1.0

Oil

Water

Oil - wet

Swc > 20 to 25% < 15%, usually 10%krw 0.5, approaching 1

Water Wet Oil Wet


Gas relative permeability remainszero until the critical gas saturationis reached(Sgc)

SL

Swc

Sgc

kr

1

10

Sorg

gasoil

0

Gas relative permeability will reach amaximum value at connate watersaturation plus residual oil saturation

In a gas cap, residual oil may be 0 ifno oil is present in the gas cap andthe maximum is at connate watersaturation

Gas-oil relative permeability curves are similar to water-oil


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In summary, Darcys Law can be extended to multiphase flowthrough the concept of relative permeability.

For each phase the absolute permeability, k, is reduced bymultiplying it by the appropriate relative permeability factor(between 0 & 1).

L

pA

kkq o

o

ro

o

=

L

pA

kkq w

w

rw

w

=

L

pA

kkq

g

g

rg

g

=


Relative permeability provides a quantitative measure of howeach fluid flows in relation to other fluids present.

This information is extremely valuable to the reservoir engineer whenattempting to history-match the production history of oil and gas fields.

Prior to reservoir production the only way relative permeability datacan be derived is from empirical or historical correlations, or fromlaboratory measurements.

Traditionally, laboratories generate relative permeability data usingsimplified fluid flow protocols eg at ambient conditions of temperatureand pressure, using fluids which are not fully representative of thereservoir, and/or with altered or artificial wettability.


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Inter-facial tension (IFT) varies according to pressure andtemperature, and govern fluid spreading coefficients, which, inturn, determine the ultimate hydrocarbon recovery.

Wettability has a significant effect on multi-phase flow through porousmedia and relative permeability curves and residual oil saturations, etc.,are dependent on it.

Due to the complex equipment set-up required, three-phase testing israrely performed, and almost never at reservoir conditions. However,numerous flow phenomena may be characterised by results from

standard two-phase flow tests.

Ideally three phase relative permeability data would be available fordetermining production characteristics of more complex flow scenarios.Generally forced to rely on Stones Correlations which attempt togenerate three-phase data from two-phase tests.

Reservoir EngineeringFundamentals

Semester 1 2009Semester 1 2009Semester 1 2009Semester 1 2009

Master of Petroleum Well Engineering

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Lecture 4 (Ch5) FunRE 2009