SLP Archie

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SUGAR LAND

LEARNING CENTER

Archie’s Equation

SELF - LEARNING COURSE

USEFUL PRE-REQUISITES

A Knowledge of the drilling process.



Archie’s Equation Self Learning Package

Sugar Land Learning Center Page 2





INTRODUCTION TO SELF-LEARNING

Self-learning enables you to learn at your pace, in your time and in your way. This course book provides

the content, structure and organization of your learning, which would otherwise be managed by an

instructor in a class. It even asks you questions to help you to confirm your understanding - as they

probably would! So you need to work through the document and here is some information to help you.

Layout the document is laid out consistently as shown opposite. You will see each time you turn a page: -

- The text to follow on the right hand page

- A header on this page so you know where you are

- Text headings, side headings and in-text sub-headings as shown

- The diagrams on the left hand page - referred to in the text

- The self-test questions below the line on the right- The answers to the questions on the next page, also below the line

- Page numbers

How to use the course book

- Try to do the course in a maximum of 2 sessions, close together: (e.g. Work session : break

: Work session)- Set yourself up in a suitable environment: no noise, no interruptions, etc.

- Use the self-test questions to confirm that you have understood. They are yours; they are not

assessed or marked. If you get an answer wrong, just go back through the material. Self -test

questions will be drawn only from the page on which they appear

- It is useful, but not essential for you to have met the pre-course requirements stated on thecourse book cover.





Table of Contents

OBJECTIVES.........................................................................................................................................6

INTRODUCTION...................................................................................................................................7

THE APPARATUS OF THE EXPERIMENT ....................................................................................................9EXPERIMENT 1: CUBE CONTAINS ONLY FORMATION (SALT) WATER ........................................................11EXPERIMENT 2: CUBE CONTAINS ONLY HYDROCARBON.........................................................................13EXPERIMENT 3: CUBE CONTAINS ONLY SAND........................................................................................15

EXPERIMENT 4: CUBE CONTAINS SAND AND SALT WATER .....................................................................17

EXPERIMENT 5: CUBE CONTAINS SAND AND SALT WATER WITH MORE SALT ...........................................19

EXPERIMENT 7. A MIXTURE OF 50% WATER AND 50% HYDROCARBONS ARE ADDED TO DRY SAND .........21

ARCHIE’S EQUATION.............. ............. ............. ............. ............. ............. ............. ............. ............. .23

NUMERICAL EXAMPLES OF ARCHIES EQUATION.....................................................................................25

EXPERIMENT 6: IRREDUCIBLE WATER. ..................................................................................................27

EXPERIMENT 7: IRREDUCIBLE WATER AND OIL MIXTURE........................................................................29DETERMINING RW................................................................................................................................33

READING A LOG SCALE ........................................................................................................................35

RO /RT QUICK LOOK.............................................................................................................................37

END OF ARCHIE’S EQUATION SELF LEARNING PACKAGE ............. ............. ............. .............39

GLOSSARY OF TERMS USED...........................................................................................................39





SUGAR LAND LEARNING CENTER

ARCHIE’S EQUATION SELF LEARNING

PACAKAGE

Welcome to the Archie’s equation self learning self-learning course.

Archie’s equation represents an important building block in the development of quantitative analysis of a

set of LWD or wireline logs. Therefore it is important to understand how the equation was developed

through simple experiment and how it may be extended to fit different rock type and structure.

Everything you need to complete the self-learning course is contained in these notes.

To start this self-learning course, turn to the next page.





Objectives

Upon completion of this training module, you should be able to:

• Describe Archie’s experimental setup.

• Describe the effect of filling the experimental cube with salt water.

• Describe the effect of filling the experimental cube with oil.

• Describe the effect of filling the experimental cube with sand.

• Describe the effect of filling the experimental cube with sand and salt water.

• Describe the effect of filling the experimental cube with sand and salt water with extra

salt.

• Describe the effect adding a 50-50 mixture of salt-water and oil to the sand filled cube.• Write down a simple form of Archie’s equation.

• Perform calculations using Archie’s equation.

• Describe what irreducible water is.

• Memorize the form of Archie’s equation for Sandstone and for Limestone

• Describe how and where Rw is obtained from a log.

• Demonstrate how to read a logarithmic scale.

• Perform Ro/Rt quick look method calculations.





Introduction

We can make measurements and collect data all we want, but it is of no use to us if we cannot make sense

of the measurements. The basis of all initial interpretations of wireline logging data and logging data

collected while drilling is an empirically derived equation called Archie’s equation.

Archie’s equation is an empirical (derived from experiment) formula which is normally used to estimate

water saturation in a reservoir using formation resistivity, porosity and formation fluid resistivity. A cube isused as the model so that the reading is directly calibrated to resistivity. This packet explains the simplest

form of Archie’s equation. Many other formulas, which deal with more complex conditions, are derived

from this basic equation.









The Apparatus of the Experiment

Objective: Describe Archie’s experimental setup.

In this experiment we are going to see if there is a link between porosity, the resistivity of fluid within porespaces and the resistivity measured in a piece of formation.

Throughout we will consider a 1-meter cube filled with liquid, sand and oil individually or as a mixture.

We will measure the resistance from one side of the cube to the other. Because we have chosen a 1 meter

cube the resistance will be the resistivity of the materiel in the cube.

We will monitor the water saturation of the cube with the scale on the left-hand side of the cube. The

saturation is the percentage of water in the pore spaces and not the total volume of the cube.

We will monitor the porosity of the material in the cube with the meter on the right hand side this is the

amount of space in the cube available for fluid.

DefinitionsThe Cube: A conductive material 1-m long with a cross-sectional area of 1 m

2

Pore Space: The interstitial gap between rock matrix grains

Porosity: (φ) the amount of pore space divided by the total volume of the rock considered.Saturation: The amount of liquid filling the pores. 100% saturation equals 1.0

Water Saturation: (Sw) the amount of water as a percentage of the pore spaces.

Hydrocarbon Saturation: (Shr) the amount of water as a percentage of the pore spaces. If both oil and

water were present then Sw + So = 1.

Resistance: (r) the opposition of a substance to the flow of electrical current. It describes a specific object,

can be measured directly, and is expressed in ohms.Resistivity: (R) a physical property of a substance. Resistivity is the resistance of a conductive material of

a given length and cross sectional area.

______________________________________________________________________________________

Q1 If the porosity is 50% and the water saturation is 50% what is the volume of the rock and of the

water, in cubic meters, contained within the cube?





______________________________________________________________________________________

A1 0.5M3 of rock and 0.25M

3 of water (0.25M

3 of unoccupied space).





Experiment 1: Cube contains only formation (salt) water

Objective: Describe the effect of filling the cube with salt water.

In this first experiment we fill the cube with just salt water.

The porosity, φ, is 100% because there is no rock in the cube.

The water saturation is Sw = 1.00. (100%) because all the available pore spaces is filled with water.

The hydrocarbon saturation is Shr = 0.00 (0%) because there is no hydrocarbon in any of the available

space.

Clearly whatever the resistance of the liquid we place in the cube will be the resistivity measured by the

meter because the cube is entirely water. This resistivity value, the resistivity of the water is termed Rw, the

formation water resistivity.

________________________________________________________________________Q2 If the cube is filled with any non-porous material why will we always read the resistivity of the

material?





______________________________________________________________________________________

A2 Because the cube has dimensions of 1 cubic meter

ohmmm

mohms=

× 2





Experiment 2: Cube contains only Hydrocarbon

Objective: Describe the effect of filling the cube with oil.

In this second experiment we fill the cube with just hydrocarbon oil for example.

The porosity, φ, is 100% because there is no rock in the cube.

The water saturation is Sw = 0. (0%) because all the available pore spaces is filled with oil.

The hydrocarbon saturation is Shr = 1.00 (100%) because there is only hydrocarbon in the available space.

Now the meter will read an infinitely high resistivity because oil is non-conductive.

______________________________________________________________________________________

Q3 What would the measured resistivity be if the cube were filled with pure water and why?





______________________________________________________________________________________A3 It would be very large (infinity) because pure water contains no dissolved ions, which is what

makes salt water conductive.





Experiment 3: Cube contains only Sand

Objective: Describe the effect of filling the cube with sand.

In this third experiment we fill the cube with just sand.

The porosity, φ, is approximately 30% because the grains of sand are approximately spherical and

regardless of how well we shake the cube to distribute the sand it can only fill 70% of the available volume.

This means that 30% is available to be filled with liquid before overflowing the cube, but remember in thisexperiment there is only sand ands no liquid in the cube.

The water saturation is Sw = 0. (0%) because there is no water in the cube at all.

The hydrocarbon saturation is Shr = 0.00 (0%) because there is no hydrocarbon in the cube.

Now the meter will read an infinitely high resistivity because sand is non-conductive.

______________________________________________________________________________________

Q4 Will all types of rock be non-conductive?





______________________________________________________________________________________

A4 No, clay is conductive and many metallic ores, pyrite (a copper ore) for example.





Experiment 4: Cube contains Sand and Salt water

Objective: Describe the effect of filling the cube with sand and salt water.

In this forth experiment we add salt water to the sand filled cube.

The porosity remains at 30% because porosity is a measure of the space available to be filled by fluid. This

has not changed by adding salt water.

The water saturation is Sw 1.0 (100%) because all the available pore space is filled with water.


Now the meter will read a value of water filled rock resistivity called Ro. The value will be higher than Rw

because that is the value of the salt water in the pore spaces but less that infinity because electricity willfind a path through the pores from one side of the cube to the other.

______________________________________________________________________________________

Q5 The volume of a sphere is 4/3πr3. What is the porosity if the cube is filled with one ball of radius

0.50 meters? And what is the porosity if it is filled with a million balls of radius 0.005 meters?





______________________________________________________________________________________

A6 47.6%.





Experiment 5: Cube contains Sand and Salt water with more salt

Objective: Describe the effect of filling the cube with sand and salt water with more salt.

In this fifth experiment we add more salt to the salt water in the sand filled cube and see what happens to

Ro.

The porosity remains at 30% because porosity is a measure of the space available to be filled by fluid. This

has not changed by adding salt water.

The water saturation is Sw 1.0 (100%) because all the available pore space is filled with water.


The meter will read a lower value of resistivity than in experiment 4 because the extra salt adds more ionsto the water.

The reading is still considered to be Ro but is different because Rw is different.

Now the meter will read a value of water filled rock resistivity called Ro. The value will be higher than Rw

because that is the value of the salt water in the pore spaces but less that infinity because electricity willfind a path through the pores from one side eof the cube to the other.

______________________________________________________________________________________

Q7 There is a limit to the amount of salt that can be dissolved, what happens when more salt is added

after the maximum has been reached?





______________________________________________________________________________________A7 The salt will exist as solid salt within the liquid. Solid salt is non-conductive.





Experiment 6. A mixture of 50% water and 50% hydrocarbons are added to dry sand

Objective: Describe the effect of adding a 50-50 mixture of salt-water and oil to the sandfilled cube.

In this experiment a fifty-fifty mixture of oil and water is added to fresh dry sand.

The porosity is 30% because the same sand is used.

The water saturation is Sw = 0.50 (50%) because half the available pore space is filled with water.

The hydrocarbon saturation is Shr = 0.50 (50%) because half the available pore space is filled with oil.

Now the meter will read a higher Rt than if the pore space was just filled with water

______________________________________________________________________________________

Q8 What would happen to the value of measured resistivity if the percentage of oil were higher?





______________________________________________________________________________________A8 It would be higher because more space is taken up by non-conductive oil.





Archie’s Equation

Objective: Write down a simple form of Archie’s equation.

Archie continued making measurements of the cube with many different values of Rw and porosity rock

within the cube.

He was able to derive an equation that gave a close approximation to the readings he was seeing on his

meter. This equation is given below.

We will discover later how this equation is modified for various formation conditions.

The equation tells us that if saturation and porosity remain constant then Rt is proportional to Rw.

If Rw and Sw are constant then Rt varies inversely with the square of porosity. In other words an increase

in measured resistivity may be due to a reduction in formation porosity.

______________________________________________________________________________________

Q9 Rw is the formation water resistivity, what happens to it with increasing temperature?

22Sw

Rw Rt

×=

φ





______________________________________________________________________________________

A9 The resistivity of a solution of dissolved salt decreases with increasing temperature. Look at chartGen-9 in the Schlumberger chart book.





Numerical examples of Archies equation.

Objective: Perform calculations using Archie’s equation.

Let us consider some numerical examples.

Remember

1. If Rw = 0.5 Ohmm, Sw = 1.00 (100%) & Por = 0.3 (30%)

Rt = 5.56 Ohmm

2. If Rw = 0.5 Ohmm, Sw = 1.0 (100%) & Por = 0.2 (20%)

Rt = 12.5 Ohmm

3. If Rw = 0.25 Ohmm, Sw = 1.0 (100%) & Por = 0.2 (20%)

Rt = 6.25 ohmm

______________________________________________________________________________________Q10 If Rw = 0.2 Ohmm, Por = 33% & Rt = 7 Ohmm what is Sw?

22Sw

Rw Rt

×=

φ

22 00.13.0

5.0

×= Rt

22 00.12.0

5.0

×= Rt

22 00.12.0

25.0

×= Rt





______________________________________________________________________________________A10 Sw = 0.51Ohmm

Rockmatrix

PoreS ace

IrreducibleWater

Rt

RwSw

×=

2φ 733.0

2.02×

=Sw





Experiment 7: Irreducible water.

Objective: Describe what irreducible water is.

With the cube containing sand and salt water, the water is drained from the bottom of the cube. However

when no more water will drain from the cube some water still remains in the cube. This water is actually

bonded to the sand grains and cannot be removed; this water is called the irreducible water saturation.

The lower diagram shows a blown up cross section of the sand grains after the water has been drained off.

Some of the water remains in place due to a property know as adhesion. Adhesion is the attraction of

unlike elemental bodies. The water molecule H2O has an imbalance in its physical structure that gives it a

charge imbalance or polarity. This quality is what causes some of the water to remain stuck to the surface

of the rock grains.

______________________________________________________________________________________

Q11 Is the amount of irreducible water likely to vary with grain size?





______________________________________________________________________________________

A11 In a given volume, smaller grains will have a larger total surface area than bigger grains.

RockMatrix

Oil in Pore

S ace

WaterCoatin





Experiment 8: Irreducible water and oil mixture.

Objective: Perform calculations using Archie’s equation.

With the knowledge of irreducible water lets go back and look closely at the sand grains while there is afifty-fifty mixture of oil and water in the pore spaces.

A cross-section of the grains revels the water adhering to the grain surfaces while the hydrocarbon is not. If

we were to drain off the liquid in the cube we would recover all of the oil and most of the water again

leaving some water around the rock grains.

All natural hydrocarbon reservoirs contain some amount of water bound to the rock and often an amount of

water not that is free as well as the hydrocarbon. In other words the hydrocarbon saturation will never be

100% and in fact is rarely over 90%.

______________________________________________________________________________________Q12 Will bound water affect my resistivity measurements?





______________________________________________________________________________________

A12 Yes, because the water is conductive.





Updating Archie’s Equation

Objective: Memorize the form of Archie’s equation for Sandstone and for Limestone.

Since Archie’s Law was developed in 1942, petrophysicists have tested his equation by repeating hisexperiments for almost every conceivable type of rock: igneous and metamorphic as well as sedimentary.

While Archie’s equation held true for most formation types, their experiments showed that Archie’s

formation factor needed to be updated to take into account not just porosity, but other variables as well:cementation, tortuosity, granular shape and size.

a = local correction factor, takes into account a myriad of variables that can affect a formation in

a given location. It is typically equal to 1 for carbonate rocks and 0.62 for sands.

m = cementation factor, represents how the grains in a formation are held together and also varies

based on local formation characteristics. It is typically equal to 2 for carbonate rocks and

2.15 for sands.

n = saturation exponent. It is typically equal to 2.

Note: Rw will be obtained in a zone of the formation known to contain no hydrocarbons, where theporosity is known. The operating company will usually know the value of Rw in any of their fields.

The first term of Archie’s equation is often replaced with a term called formation factor, F, as derived

below.

By means of a graph, the "Porosity vs. Formation Factor" figure relates porosity to the formation factor for

several known formations. This demonstrates that the approximations only diverge at lower porosity.

The measurement of resistivity is influenced not only by the fluid in the reservoir and the porosity but also

by clay particles if present in the pore spaces. There are a number of methods and equations that improvethe basic Archie equation to allow for the influence of clay on our measurements. These methods are

beyond the scope of this package.

______________________________________________________________________________________

Q13 What is the expression of Archie’s equation in Sandstone and then in Limestone?

nmSw

Rwa Rt ×=

φ

F am =

φ





______________________________________________________________________________________

A13

22

1

Sw

Rw Rt Limestone ×=⇒

φ 215.2

62.0

Sw

Rw Rt Sandstone ×=⇒

φ





Determining Rw

Objective: Describe how and where Rw is obtained from a log.

Central to the determination of the saturation of the reservoir is determining Rw, the resistivity of theformation water. This must be read from a resistivity and porosity log through zone of 100% watersaturation or by determination from cores. We deal with determination of Rw from resistivity logs.

First we must drill through a water zone and log it with a resistivity tool. Each tool has various

characteristics depending of the environment these must be corrected for. Since this is tool specific we it

will not be dealt with here.

______________________________________________________________________________________Q14 What is Rw?





______________________________________________________________________________________A14 Rw is the resistivity of the water in the pore spaces.

0.1 1 10 100 1000

1 2 3 4 5 6 7 8910 20 30 40 5060 80100

1 2 3





Reading a Log Scale

Objective: Demonstrate how to read a logarithmic scale.

Resistivity logs are typically presented on logarithmic scales. These scales are different from regular linear

scales in that they show a decade or power of ten for each division. For example in the upper plot the left-

hand scale starts at 0.1 and finishes at 1000. Each step is a power of ten or decade so the plot opposite is a

four-decade plot.

The problem with this type of plot is that the individual dividers between the major points are not spacedequally as the middle diagram demonstrates.

Resistivity logs are usually presented from 0.2 to 2000 on a 4-decade log scale with depth, as in the lower

diagram, although 0.2 to 200 is also used.

Point 1 is a point with a value of 0.34 and NOT 0.35 because the dividing lines are not equally spaced.

______________________________________________________________________________________

Q15 What is the value of point 2 and 3?





______________________________________________________________________________________A15 2 = 2.4 ohmm 3 = 20 ohmm

A

B

C

D





Ro/Rt Quick Look

Objective: Perform Ro/Rt quick look method calculations.

There is a way of estimating water saturation and therefore hydrocarbon satruration called the Ro/Rt quick

look method that requires no porosity measurement. It assumes that the reservoir has a constant porosity.

If you recall the development of Archie’s equation which allows for any rock type etc. equation (1).

We can define a value called Ro which is the resistivity read in a zone of 100% saturation. This leads to

equation (2). Which can be rearranged in termns of the Formation factor ging equation (3).

If we now rewrite equation (1) in terms of formation factor we get equation (4).

Equating equations (3) & (4) we arrive at equation (5) which completely eliminates the formation factor

term aslong as the porosity is constant.

We can then write the equation in terms of Saturation, equation (6).

And more generally when n = 2 we are left with equation (7). This now gives us a quick look ratio maethod

to asses the saturation.

From the log opposite we choose our lowest resistivity in a clean part of the reservoir, point A, and read

the value; 0.6 ohmm. This is Ro the value of resistivity for 100% water saturation.

So for point B where Rt = 4 ohmm:

______________________________________________________________________________________Q16 What is the value of Hydrocarbon saturation at points C and D?

( )2.......

1

Rwa Ro

m ×=

φ

( )1.......nm

Sw

Rwa Rt ×=

φ

( )4.......m

na

Rw

Sw Rt

φ =

×

( )3.......

Rw

Roa

m =

φ

( )5....... Rw

Ro

Rw

Sw Rt n

=×

( )6.......n

Rt

RoSw =

( )7....... Rt

RoSw =

38.04

6.0==Sw = 38% water saturation which means 62% hydrocarbon saturation.





______________________________________________________________________________________A16 Point C Rt = 5.3

Point D Rt = 4.5

34.03.5


37.05.4





End of Archie’s Equation Self Learning Package

Congratulations you have successfully completed the Archie’s Equation self-learning package. Take the

time now to go back to the beginning of the document and review the learning objectives

Glossary of terms Used

Formation Matrix The solid rock part that makes up the formation

Limestone Calcium Carbonate deposits often from the shells of sea animals.

Permeability The measure of how well a fluid can flow though the pore spaces of the formation.

Pore Gap/space between rock grains/particles, usually filled with a liquid either water or

hydrocarbon.

Porosity The ratio of solid rock to pore space per unit volume of formation.

Ra Apparent resistivity, the resistivity measured by the tool.

Rm Resistivity of mud.

Rmf Resistivity of mud filtrate.Ro Resistivity of 100% water saturated zone.Rs Resistivity of the adjacent bed.

Rt Resistivity of “virgin” uninvaded zone.

Rw Resistivity of formation water.

Rxo Resistivity of fully flushed zone.

Shale Sedimentary rock made up of fine particles.

Sw Formation water saturation.

Sxo Flushed zone water saturation.

tmc Thickness of mud cake.

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