2a Porosity Permeability

7/29/2019 2a Porosity Permeability

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Single phase flow in porousmedia: Darcys law


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Oil or gas reservoir

Limestone reservoir Sandstone reservoir


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Vp

Vr

rp

p

VVVV

V

Usually = 0.05 0.40

Porosity


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Rock Matrix and Pore Space

Rock matrix Pore space


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Pore Structure

Typical Pore Structure


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Scanning Electron Micrograph

Norphlet Formation, Offshore Alabama, USA

Pores Provide the

Volume to ContainHydrocarbon Fluids

Pore Throats Restrict

Fluid Flow

Pore

Throat

Porosity in Sandstone


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Porosity of common rock types


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Permeability

Permeability k [D, mD]

-capacity of rock to transmit fluid

- function of open space and its interconnection

- depends on properties of rock formation

Permeability


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Permeability (Darcys Law)

Darcys experiment was performed to design a filter large enough

to ensure the daily requirement of water for the city of Dijon(1856).

1 2h hq KA

L

q: volumetric rate [m3/s]K: hydraulic conductivity [m/s]A: Cross-sectional area of sandpack [m2]h: piezometric head [m]L: length of sand pack [m]


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Permeability (Darcys Law)

In Petroleum Engineering we use phase potentials

p gh

KA pq g

g L

kA pq g

L

kis permeability and property of a

rockUsually expressed in D or mD (Dstands for Darcy)1 Darcy = 10-12 m2 = 1 m2


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Darcys Law and dip angle

sinkA pq gL

zk

u p ge u is Darcys velocity

,x y

z

k p k pu u

x y

k pu g

z


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Definition of parameters

Total flow rate = total discharge [m3

s-1

]: Q

Darcy velocity u = specific discharge q [m s-1]:

Interstitial velocity = linear velocity or pore velocity v:

Q

u A

uv


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Definition of parameters

Hydraulic gradient: In empirical Darcys law; ratio ofdifference in piezometric head (P/g + z) and length of the

sand pack; sand pack arranged in vertical position

Potential gradient: Analogous to hydraulic gradient;sand pack position not restricted to vertical position; inmore generalized Darcys law; ratio of difference in fluidpotential and length of the sand pack

2 1h h

L

2 1

L


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Bundle of capillary tubes

Many pore space models are based on capillarytube bundle

2

2

8

r pq r gL

Hagen-Poiseuille law for laminar flow:

kA pq g

L

Darcys law:


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Restrictions and assumptions ofDarcys law

- Laminar flow

- For Reynolds numbers between 1 and 10

- No inertial forces

- Viscous forces predominant

- No slip (zero velocity of fluid at wall)

- Incompressible fluid (=constant density)

- Viscosity of water

Application to gas flow through porous medium not

appropriate

Note: Transition to turbulent flow for Re between 60 and 150.


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Permeability

Permeability is influenced by:

Pore size and pore-size distribution

Grain size

Grain-distribution

Compaction (which is function of pressure)

Grain shape

Klinkenberg effect


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Permeability measurementmethods

core scale

Inject a fluid with defined properties

Use Darcys law to calculate permeability

Well test

Measure flow and pressure

Calculate permeability


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Representative ElementaryVolume (REV)

To use equations we need average values for permeability,

saturations, porosity, over a volume The volume must be small with respect to our problem of interest

and large enough such that the averaged quantity does notchange significantly if we increase the averaging volume by afactor of say two.


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Darcys law and Navier-Stokes equations

NS is continuum form of Newtons second law

inertia pres grav visc

visc

S

S V

V V V V

F F F F

dm pdA m F

dt

pdA pdV

ddV pdV dV dV

dt k

vg

v ug

We neglect inertia forces:

0

V V V

pdV dV dVk

ug

kp

u g


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Limitations of applicability ofDarcys law

- Due to skin formation permeability in vicinity ofwellbore is changing (decreasing)

- At higher flow rates inertial force, acting due toconvective acceleration of fluid particles throughporous medium, have to be taken into account


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Forchheimer equation

- Modification of Darcys law taking the inertial forces into

account- Inertial forces need to be attributed for Re numbershigher than 10.- In porous media inertial forces need to be taken into

account because of acceleration and decelerations of fluidparticles through pore spaces NOT because of turbulenceflow.- Originally derived for flow of fluids through pipes whereat high velocity distinct transition from laminar to turbulent

flow- Additional pressure drop due to skin formation can bedetermined applying Forchheimer equation.


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Forchheimer equation is a phenomenological approach.It was recognized that Darcys law deviates for Re numbersof around 10. To correct for this the inertial forces werealso taken into account. These forces describe the fact thatin porous media the fluid flow is accelerating or

decelerating due to the tortuosity.The Forchheimer equation was stated on this pictureempirically

Herein is theinertial parameter.

2u u

L k


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As common for phenomenological approaches, thephenomenological parameters are related to measurableparameters by correlations. and can be related to porestructure parameters by:

With the constantsA=180 and B=1.8

In general, 1/kand the inertia parameter can bededuced from experimental data on the drop of thepiezometric head as function of the Darcy velocity.

2

23 3

1 11

P PA D B Dk


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Permeability-porosity correlations

General form of correlations:

k = Shape factor * Porosity factor * square of grain sizediameter

3 2

2

1

72 1

pD

k

Carman-Kozeny correlation:

Tortuosityis a variable that defines the straightness ofthe flow paths


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Other Data Used in Well Testing


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Formation Volume Factor

surfreso

V

VB


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Viscosity

dyv

v + dv

dy

dvA

F


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Fluid Compressibility

p

Vln

p

V

V

1

coo

oo


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Pore Compressibility

p

ln

p

1

cf


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Net Pay Thickness

h = h1 + h2 + h3

Shale

Sand

h3

h2

h1


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Net Pay Thickness

Case 1 Case 3

Case 4Case 2


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Wellbore Radius

rw


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Total Compressibility

ggwwooft cScScScc


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Radial (steady state) Darcys Law

qBrw

h

Radial system in steady state

rrw re

pw

pe

p

qB

qB

pe

pr

Permeability can be derived:

Note: B assumed constant

p ur k

2

qBu

rh

2

qB dr dp

kh r

ln2

w

w

qB rp p

kh r

2

ln /w

w

khq p p

B r r

ln /

2

w

w

qB r r k

h p p


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SkinSkin is any near wellbore phenomenon that causes an additionalpressure drop extra to that expected from Darcy inflow (Delta P-skin),e.g. damaged rock:

Stimulated

Less Pressure drop

due to hydraulic frac

Can be negative too

!

Undamaged Damaged

Extra Pressure drop

due to damage skinpskinExpected flowing pressure

undamaged

Actual flowing pressure

Positive Skin: drilling mud filtrates, clay swelling, mechanically

destroyed rock, gravel packNegative skin: acid jobs, extra deep perforations, hydraulically fractured


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Skin and Productivity Index

2skin

qp S

kh

van Everdingen equation:

ln2

ee wf

w

rqp p S

kh r

e wf

qPI

p p

Productivity Index:

Ways to improve PI:

Skin removal Increasing effective permeability Viscosity reduction Reduction of Bo Increasing well penetration h


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Geometric Skin

Partially perforatedFully perforated

Pressure drop dueto geometric skin


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Flow in parallel

n

j

j

n

j

jj

h

hk

k

1

1

Linear Radial

h1h1

h2

h3

h2

h3

k1k2

k3

Q1Q2

Q3

QTre

Q1Q2

Q3

QT

P1

P2 PePwhT

L


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Flow in series

Linear

Often used for

vertical permeability

k1 k2 k3

h

P1 P2

Q Q

Radial

re

Q

PePwL1

L

L3L2

r2

r1P1 P2 P3

n

j j

j

k

LLk

1


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Permeability averaging

Parallel Flow

Series Flow

Random Flow

ArithmeticAverage

Harmonic

Average

Geometric

Average

n

j

j

n

j

jj

h

hk

k

1

1

n

j j

j

n

j

j

k

h

h

k

1

1

nnkkkk

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2a Porosity Permeability