What is the cementation exponent? A new differential...

EGU2009-319

What is the cementation t?exponent?

A new differential interpretationA new differential interpretation

Paul GloverUniversité Laval Québec CanadaUniversité Laval, Québec, Canada

PlanPlan

Plan

Introduction – The power of Archie!What is the cementation exponent?

Introduction

What is the Cementation

exponent?

Traditional interpretationsA new differential interpretationexponent?

Traditionalinterpretations

A new

pConductivity regimeConnectedness and connectivityDifferential forminterpretation

Conclusions

Acknowledg-ments

Differential form

Conclusionsments

IntroductionGl b l h d b d tiGlobal hydrocarbon production

OilOil

Plan

Discoveriesin 2003

182,000,000,000 bbl.

4,500,000,000,000US$

Introduction


exponent?

in 2003 US$

Oil & GasJohnson et al., 2004

exponent?


A new

Between 1950 and

1,500,000,000,000 bbl. oil

7.5 Tscf gasOil & Gas

interpretation

Conclusions

Acknowledg-ments

2002,

Bentley, 2002ments Over half of these resources has already been produced,

and has driven the global economy for the last fifty years.

IntroductionTh f A hi !The power of Archie!

Discoveries made using some expertise in g p• Geology, Geophysics, Engineering,

andPlain Good Luck

Plan• Plain Good Luck

Introduction


exponent?

HoweverALL reserves calculations were made usingexponent?


A new

ALL reserves calculations were made using petrophysics measurements and Archie`s equations

interpretation

Conclusions

Acknowledg-ments

It is difficult to overestimate the impact of either the petrophysical techniques or Archie`sments the petrophysical techniques or Archie s relationships on the worldwide economy.

What is the t ti t?cementation exponent?

Plan

Archie`s laws link the electrical resistivity to porosity, the resistivity of the pore water, and to the fractional saturation of the pore space with the water.

Introduction


exponent?

saturation of the pore space with the water.

Used to calculate the hydrocarbon saturation of the reservoir rock hence the hydrocarbon reserves. exponent?


A new

y

Contain two exponents, m and n, which Archie called the cementation exponent and the saturation

interpretation

Conclusions

Acknowledg-ments

exponent, respectively.

The conductivity of the hydrocarbon saturated rock is hi hl iti t h i ith tments highly sensitive to changes in either exponent.

What is the t ti t?cementation exponent?

Plan

The cementation exponent commonly takes values from just over 1 to around 5.

W t d il t ti l l t d ith A hi ’Introduction


exponent?

Water and oil saturations calculated with Archie’s equations are highly sensitive to this level of variability in the cementation exponent. exponent?


A new

Thankfully, there are a number of ways in which the cementation exponent can be calculated with

interpretation

Conclusions

Acknowledg-ments

precision,

which is why it has often been relegated to the status f fitti t d h h t i d tments of a fitting parameter and why no one has tried to

understand its physical meaning.

Traditional i t t tiinterpretations

ρ ( )log F

Plan

mF −= φo

w

F ρρ

= ( )( )

loglog

Fm

φ= − m

o waρ ρ φ−=Archie’s False ‘a’

Resistivityformationfactor F

Introduction


exponent?

first law

I t t ti f

Practicaldefinition

exponent?


A new

Interpretations of m(1) A factor related to the cementation of the rock (Archie, 1942).(2) Something to do with the degree of connection of the pores.

interpretation

Conclusions

Acknowledg-ments

(2) Something to do with the degree of connection of the pores.(3) A fitting exponent in an empirical relationship. (4) Only analytically defined for tubes (m=1) and spheres (m=1.5).

ments(5) The power of a fully analytical equation (Ewing and Hunt, 2006).(6) Minus the gradient of F/φ relationship in log-log space.

Conductivity regime

Need to use conductivity in place of resistivityPlan

Not trivial – but fundamental

We use resistivity for purely traditional reasons (S hl b b 1927)

Introduction


exponent? (Schlumberger bros., 1927)

However, conductivity has better physics pedigree

exponent?


A new

whereinterpretation

Conclusions

Acknowledg-ments

n qσ β=gradVσ σ= = −J Ements

Connectedness I

Now define a conductivity formation f t 1factor, G

Plan

1o

w

GF

σσ

≡ =G, like F, is also approximately constant for a given facies

Introduction


exponent?

constant for a given facies.

The conductivity formation factor varies from zero, which represents the case where (i e when0σ = 0φ →

o wσ σ=

exponent?


A new

which represents the case where (i.e., when ) and increases as the porosity increases, with (i.e., ) as .

0oσ 0φ →1G →

o wσ σ= 1φ →interpretation

Conclusions

Acknowledg-ments

G is the conductivity of the rock normalised to the conductivity of the saturating fluid.

ments G describes the conductivity of a solid/fluid mixture relative to a sample composed only of the fluid.

Connectedness II

G is a dilution factor where the pore fluid is diluted by p yrock grains.

G is a dilution factor where the conductivity of the rock Plan

is not only affected by the replacement of a given volume of fluid with the same volume of solid matrix, but also by the arrangement of the resulting solid matrix.

Introduction


exponent?

o wσ σ=

also by the arrangement of the resulting solid matrix.

Hence, G is also a measure of the availability of pathways for electrical transport.

exponent?


A new p y p

G is, infact, a measure of connectedness of the pore and fracture network of a sample.

interpretation

Conclusions

Acknowledg-ments

Hence we will define G to be the connectedness of a porous medium.

ments

Connectivity I

A f A hi ’ fi l i h d i iA restatement of Archie’s first law in the conductivity regime uses the following relationships

Plan

Introduction


exponent?

mG φ= ( )( )

loglog

Gm

φ= m

o wσ σ φ=exponent?


A new

No better physical interpretation of m than their equivalents in the resistivity regime.

interpretation

Conclusions

Acknowledg-ments

q y g

If we define a connectivity hence1χτ

≡ 1mχ φ −=ments

Connectivity II

Hence the connectedness becomes

Plan1m mG φ φφ φχ−= = =

Introduction


exponent?

The connectedness G of a rock is due to exponent?


A new

(1)The amount of pore volume available for electrical conduction (porosity φ), and

(2)Th h h i i d i hinterpretation

Conclusions

Acknowledg-ments

(2)The way that that porosity is arranged in three dimensions (represented by the connectivity χ)

ments

Differential form

The rate of change of connectedness with porosity is

Plan1mdG φIntroduction


exponent?

1mdG m md

φ χφ

−= =

exponent?


A new

The rate of change of connectedness with porosity and connectivity is

interpretation

Conclusions

Acknowledg-ments

2d Gmd dχ φ

=ments d dχ φ

Cementation exponent Ph i l i– Physical meaning

Plan

The connectedness describes how the conductivity of 100% fluid is modified by the presence of solid non-conducting grains

The cementation exponent is the

Introduction


exponent?

conducting grains.

psensitivity of the connectedness to

changes of connectivity and

exponent?


A new g yporosity.

interpretation

Conclusions

Acknowledg-ments 2d Gments 2d Gm

d dχ φ=

Connectivity/porosity l ti hi Irelationship I

( )ddG χφ

Plan

Differentiating as a product ( )ddG md d

χφχ

φ φ= =

( )Introduction


exponent?

gives ( )1mdd

χχφ φ

−=

exponent?


A newThe rate of change of connectivity of a rock with porosity depends uponinterpretation

Conclusions

Acknowledg-ments

porosity depends upon (1) its initial connectivity, (2) the cementation exponent, and (3) th i iti l itments (3) the initial porosity

Connectivity/porosity l ti hi IIrelationship II

Plan A large initial A large initial porosityIntroduction


exponent?

connectivity will augment the change in

connectivity

has the effect of diminishing the change

in connectivityexponent?


A new

connectivity in connectivityIf you add a link between pores or cracks in a well connected pore network the result is that the

If you add a crack to an otherwise low porosity rock the connectivity will change more abruptly than

interpretation

Conclusions

Acknowledg-ments

network the result is that the network increases its connectivity more than if the same link were added to a low connectivity

t k

will change more abruptly than adding the same crack to a rock that already has a high porosity.

ments network.

Conclusions I

Connectedness G of a porous medium is defined as the availability of pathways for transport.

Connectedness G is the inverse of the formation i ti it f t th d ti it f ti f t

Plan

resistivity factor, or the conductivity formation factor.

Connectivity χ is the measure of how the pore space is arranged

Introduction


exponent? is arranged.

Connectivity χ is given by and depends upon the porosity and the cementation exponent m

exponent?


A new

1mχ φ −=upon the porosity and the cementation exponent m.

Connectedness G depends upon the t f ( it φ) d th

interpretation

Conclusions

Acknowledg-ments

G φχ=amount of pore space (porosity φ) and the arrangement of the pore space (connectivity χ).

ments

Conclusions II

The rate of change of connectedness dGwith porosity depends upon the connectivity χ and the cementation

exponent m.Plan

dG md

χφ

=p

The rate of change of the connectedness with porosity and

Introduction


exponent?d dG m⎛ ⎞

=⎜ ⎟connectivity is equal to the cementation exponent,

exponent?


A new

md dχ φ⎜ ⎟

⎝ ⎠

Hence, the cementation exponent is interpreted as being the rate of change of the connectedness with porosity and

interpretation

Conclusions

Acknowledg-ments

2d Gmd dχ φ

=p yconnectivity.

ments d dχ φ

Acknowledgments

G.E. ArchiePlan

NSERC CanadaIntroduction


exponent?exponent?


A newinterpretation

Conclusions

Acknowledg-mentsments

What is the cementation exponent? A new differential...

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