full version Archie Equation Petrophysics Essay

Archie Equation Petrophysics

Category: Science

Autor: monika 07 March 2010

Words: 5990 | Pages: 24


Archie Unleashed is an attempt to put the basic log analysis methodology for computing water saturation into a readable reference document. The beginning log analyst or petrophysicist should have little difficulty with the terms and concepts utilized in this paper, however, most terms are redefined in appendix A.

The basic outline of this document closely follows a previous work written for the casual interpeter in log analysis. Archie Unleashed is meant to carry that work one step further. Basic concepts are explained along with more detailed examples and explanations.

The personal computer has revolutionized the way we work and play. The kind and amount of data we work with in petrophysics can be easily handled in spread sheet programs that are under the full control of the user. The second section of Archie Unleashed examines several data sets and techniques for handling these data and defines a simple spreadsheet to do the calculations.

Section 1. The Archie Equation

-water saturation as a decimal fraction.
-Resistivity of the 100% water saturated rock.
-Resistivity of the rock-fluid system.

Equation 1.1 was empirically derived by G. E. Archie while working for Shell. This work was reported in in his famous 1942 paper. He plotted SW versus the ratio Rt/Ro (the Resistivity Index) on log-log paper, see Figure 1.1. This same technique is still used today to derive a value for n from core measurements.

The absolute value of n is commonly near 2 and is generally taken as equal to 2.00 when no laboratory determination of n is available. Accepting the convention of n = 2 allows the Archie equation to be written in the following forms:

These are equivalent expressions. Proper application will yield identical answers.

Critical Sw Values

The critical value of water saturation (Sw) is commonly taken as 50%. Sw must be less than the critical value for a zone to produce oil or gas without water production. Texture and sorting are the major factors affecting the maximum Sw before water free production.

The critical value for Sw is known to vary from about 30% to 80%. Carbonates often have a critical Sw value less than 50%, whereas sandstones may have critical Sw’s well over 50%.

Predicting the critical value in low porosity carbonates, fractured carbonates, or carbonates with vugular or moldic porosity may be impossible.

Assumptions of the Archie Equation:

1. The reservoir rock is water-wet (water is the wetting fluid, the fluid in contact with the grain surfaces).
2. The reservoir rock is shale free (or the amounts are small and can be safely neglected); that is, the rock matrix is non-conductive.
3. The reservoir rock has moderate to high porosity, the higher the porosity, the better the Archie equation works.
4. The reservoir porosity is intergranular (connected) in nature.

Primary Considerations in the Archie Equation:

1. The Archie equation does not predict Sw well in low porosity rocks, particularly when porosity is less than 5%.
2. A knowledge of Sw alone may not be sufficient to predict production.
3. The Archie equation does not work well where Rw is high (fresh water). Fresh water lessens the contrast in resistivity between oil and water zones. Fresh water increases the importance or effect of conductivity along grain surfaces.
4. The Archie equation may not apply at low Sw values. In the original paper, the samples had water saturation greater than 15%.
5. n is generally greater than 2 in oil-wet reservoirs, and may reach values of 10 or more.
Guidelines For Application Of The Archie Equation:

1. The quality of the water saturation values derived from the Archie equation is no better than the quality of the well logs used.
2. If you are at wellsite, check the quality of each log run. Do not hesitate to rerun logs which do not meet quality control standards. Check especially to see that logs such as the resistivity and porosity measurements are recorded on depth with each other and with the previous surveys.
3. Choose the primary electric log as a reference. Mark intervals selected for quantitative analysis on this log. Depth. correlate the other surveys to this reference log or the Gamma ray curve from this log.
4. Select intervals with at least four feet of constant response whenever possible. Use a single representative value from the center of the zone for each porosity and resistivity measurement. This procedure minimizes thin bed and bed boundary effects on the logs.
5. Consider how to keep the interpretation simple. Analyse the most clear-cut cases first. Look for:
The obvious mineral markers (salt, anhydrite).
The cleanest, most porous, intervals.
The cleanest, water-bearing, intervals.
The cleanest, hydrocarbon-bearing, intervals.
The fluid contacts: gas/oil, oil/water.
The transition zones.

6. Tackle one unknown at a time. For example, when deriving an Rw value from logs, choose a water-bearing zone as nearly identical as possible to the hydrocarbon zone of interest. This means the two zones have the same lithology, have the same or very similar porosity values, have the same type of pore structure (do not compare a zone with moldic porosity to one with granular or fracture porosity for instance), have the same formation water chemistry, etc. Ideally, the two zones are identical except for hydrocarbon content.

Bear in mind what G. E. Archie said about his equations. "It should be remembered that the equations given are not precise and represent only approximate relationships. It is believed, however, that under favourable conditions their application falls within useful limits of accuracy.'
Section 2 The Formation Resistivity Factor

Equation. 2.1 is a definition statement. As such, it is an exact expression for F, the formation factor.

Equation 2.2 is the general form of an empirically derived equation for F. Values for a and m are derived by plotting F versus porosity, see Figure 2.1.

Equation 2.2 can be written as log F log a - m log 0 Written in logarithmic form:

It is clear that a and m are just the intercept and slope of the line on the F versus 0 plot, a and m vary with rock type. They are influenced by grain size, grain size or pore size distribution, cementing agents, cement distribution, etc. a and m can be estimated from a knowledge of rock type. Local experience is very helpful.

Satisfactory results are usually obtained by using:
a = 1.0
m = 1. 8 for sandstones
m = 2.0 for carbonates (limestone or dolomites)

The lower the porosity is, the more critical the choice of a and m becomes, see Table 2.1.
For porosity values above 20%, F is not sensitive to moderate variation in a and m.
For porosity values below 10%, F is very sensitive to variation in a and m.

Major Assumptions in the Formation Resistivity Factor:
1. The reservoir is water wet.
2. The reservoir has moderate porosity.
3. The rock matrix is non-conductive.
Major Cautions for the Formation Resistivity Factor:
1. It is difficult to estimate an accurate F value for low porosity rocks.
2. Shaly sands are difficult to work with.

Percent F Values
Porosity A B C D E
10 87.6 63.1 100 50.3 29.5
20 19.7 18.1 25 17.3 13.9
25 12.2 12.1 16 12.3 11
30 8.3 8.7 11.1 9.3 9
a 0.62 1 1 1.45 2.45
m 2.15 1.8 2 1.54 1.08

A = the Humble equation, based on 29 core samples.
B = general purpose sandstone equation, 'worldwide average'.
C = general purpose carbonate equation, worldwide averages.
D = general purpose sandstone equation, 981 core samples.
E = sandstone equation, lower Pliocene (Repetto Fm.) Southern California, log derived, 1575 data points.

Major Recommendations for the Formation Resistivity Factor:

1. Use: a = 1.0
m = 1.8, sandstones
m = 2.0, carbonates
When no other information is available.
2. Use local knowledge, core measurements, and experience to aid in selection of appropriate a and m values.

Section 3.0 Rw Determination

Methods of Rw Determination
1. Measured on a water sample produced from the formation and well of interest. (Be cautious, representative water samples are uncommon.)
2. Looked up in an Rw catalog.
3. Derived from logs: 3 primary methods:
3.1 SP method, water zone preferred but not required.
3.2 Φ - Rt method, water zone required.
3.3 Rxo/Rt method, water zone required.

Except for the direct measurement of Rw on water produced from the zone of interest, all methods require the assumption that the Rw value derived from logs, or catalogues, is valid for the zone of interest.

3.1 SP Method - General Comments

The SP method assumes that the SP deflection opposite a porous and permeable zone is caused solely by the contrast between the mud and formation water chemistry. The method assumes that the equation Essp = -K log aw/amf describes the relation of SP to fluid chemistry. Therefore, if the SP deflection and mud chemistry are known, the water chemistry can be derived. The equations and charts in common use assume that sodium chloride is the dominant salt in the mud and water solutions.

The major complication encountered in the Rw calculation is that the chemical activities of the fluids are expressed in terms of equivalent resistivity. The rules and charts for equivalent resistivity often obscure what is a straightforward and simple procedure.

The SP method works most satisfactorily where the sands are shale-free, water bearing, fairly conductive (R < 10 ohm-m), and thick (> 20 feet), where the mud is fresh (Rmf > 3 Rw), and where the dominant salt is NaCl.

SP Method - Specific Technique
The technique outlined below uses the 1979 or later Schlumberger Chart books; the Dresser, Gearhart, and Welex charts are more straightforward, use them if preferred. The Baker Atlas Charts are also described.
(1) SP deflection, measured on the log from the shale baseline to the maximum deflection opposite a clean sand, see Figure 3.1.
(2) Formation Temperature (FT)
(3) Rmf at FT(mud filtrate resistivity)
Method: Schlumberger
1. Calculate: Rmfeq from Rmf at FT using the appropriate rule (listed at the top of chart SP-1 in the Schlumberger Charts). Usually &quot;rule (a)-' applies and Rmfeq = .85 Rmf.
2. Enter Chart SP-1:(See Figure 3-2). Start at the bottom:
Draw a line from the SP value through the FT value and extend the line to the Rmfeq/Rweq on the right hand side of the graph. Extend this line crossing the Rmfeq post at the appropriate intersection to the Rweq post. Record this Rweq value.
3. Enter Chart SP-2 (see figure 3.3) Start on the left :
Take the above Rweq value and enter the graph at the y axis. Move horizontally to the appropriate formation temperature line (interpolate as necessary). Drop vertically from this intersection. - Read Rw (now at FT) on the x-axis scale.

Method: Baker Atlas

1. Start with the Bed thickness correction to drive SSP from SP log.
2. Go to next Chart and derive Rmf/Rweq.
3. Knowing Rmf we can calculate Rweq.
4. Use either the Metric or English system charts to determine Rw

Major Assumptions of the SP Method:
1. The SP deflection on the log is controlled entirely by the contrast between the mud filtrate and the formation water.
2. Both fluids are dominantly NaCl solutions.
3. Rw is the same in the water zone and the hydrocarbon zone.

Major Advantages of the SP Method:
1. Rw can be calculated for the zone of interest; a water zone is not required, (see Cautions).
2. SP logs are readily available (used to be).
3. Knowledge of rock type is not required.

Major Cautions for the SP Method:
1. Resistive zones reduce the SP deflection. This includes both low porosity zones and hydrocarbon bearing zones.
2. Rmf values are frequently of pocr quality.
3. Bed thickness strongly affects SP response. If you have no beds with thicknesses at least 20 feet or more then there are thickness correction charts available. The Atlas SP method begins with corrections for bed thickness.
4. The technique works best in fresh muds, Rmf > 3 Rw
5. Shale content reduces the SP deflection. Use the cleanest zone possible.

3.2 Porosity- Rt Method: General Comments
In a water zone, the Rt value is the Ro value. This allows us to equate equations 2.1 and 2.2 and take the Ro value directly from the logs, see below:

When a=1.0

Porosity-Rt Method: Specific Technique

Two variations of this method are available.
Method One: Assume values for a and m
1. a, m - by assumption, see Section 2,
Generally use:
a = 1
m = 1.8 sandstones
m = 2 carbonates
2. Ro read directly from the deep resistivity curve on the logs, see Section IV if necessary.
3. &#966; porosity value as a decimal fraction, derived from logs by appropriate techniques, see Section 5.

Calculate: by plugging in the above values

Note: Porosity must be entered as a decimal fraction. No temperature conversions are necessary. Because Ro is measured at formation temperature, the derived Rw is already at FT.

Method Two - Solve for aRw and m graphically

This is the &#966; - Rt crossplot technique commonly identified as the Pickett Plot technique. G. R. Pickett published this procedure in the November 1966 issue of JPT. It is a very forgiving technique and deserves more use than it gets.

The Pickett Plot is simple to construct. Plot corresponding Porosity and Rt values on log-log paper, see Fig. 3-4. For convenience, use Rt as the horizontal scale, and choose the scale so that the resistivity values will plot on the right-hand half of the paper.

For any given Porosity value, the left-most point or lowest Rt value is the closest approximation to the Ro or water saturated value of formation resistivity.

The diagonal line which connects the Ro values for each porosity value defines the &#1060; -Rt relation for Sw = 100%, see Fig. 3-8.

The simplicity of this technique derives from the fact that the Archie equation can be written in logarithmic terms as shown below. (where n=2)

When Sw=100%, a decimal fraction of 1.0 then:

This equation is the form of a straight line.

Plotting Rt versus o on log-log paper for water bearing points of varying porosity yields a line with slope proportional to m and and an intercept at &#1060; = 100% of aRw, (see Fig. 3-6.)

Using Rt as the horizontal scale results in the slope of the Ro line being

And m can be taken directly from the plot by defining the absolute value of m as:

Where change in x and y are the linear distances on the plot, see Fig. 3-8.

Major Assumption of the porosity-Rt Methods:

1. A 100% water-bearing zone is present and used for Rw determination.
2. Rw in the water zone is the same as in the hydrocarbon zone.
3. The rock type in the hydrocarbon zone and the water zone is constant; that is, a and m are constant.
Major Advantage of the Porosity-Rt Methods:

The techniques are forgiving. As long as a and m or aRw and m are kept the same for the water zone and the hydrocarbon zone, incorrect selection of these parameters largely cancels out. The more nearly equal the porosity values are in the water and hydrocarbon zones, the more forgiving these techniques are.

Major Cautions for the porosity Rt Methods:

1. Rock types may change between the water and hydrocarbon zones. Look for well log or other evidence of constant rock type.
2. Rw values may change between the water and hydrocarbon zones.

Step 1: Plot the corresponding Porosity and Resistivity values
Step 2: Determine the water bearing or Ro Trend by drawing a line through the lowest resistivity values for each porosity value.
Step 3: Determine the Quantity of the aRw. This is the intersection of the Ro line and 100% Porosity line.
Step 4: Provided your plot is to scale (Determine m. ‘m’ is the inverse of the true algebraic slope. Use linear units to measure the change in x and y.

3.3 Rxo/Rt Method - General Comments

This method utilizes the fact that in a water zone:

and in the flushed zone:


and solving for Rw yields:

Rxo/Rt Method:Specific Technique
1. Rmf - at formation temperature, see Section VI if necessary.
2. Rxo ~ read directly from a microresistivity log
3. Rt - read directly from a deep resistivity log

Calculate: by plugging in the above values.

Note: Rw will be the Rw value at formation temperature as long as Rmf was converted to FT.

Major Assumptions of the Rxo/Rt Method:
1. Rt is read from a water zone.
2. Complete flushing has occurred in the flushed zone.
3. Rxo is measured by the microresistivity device.
4. Rw in the water zone and in the hydrocarbon zone is the same.

Major Advantages of the Rxo/Rt Method:
1. Does not require a knowledge of a, m, or a,.
2. Often works well in low porosity, deeply invaded zones with high resistivity where the SP method fails.

Major Cautions for the Rxo/Rt Method:
1. Good Rxo values are hard to come by.
2. Rmf values reported on log headings are commonly of poor quality.
3. Rmf must be converted to FT before calculating Rw from Eqn. III-4.

Section 4 Rt Determination

In most cases, as long as the proper Rt tool was used, Rt is simply the resistivity value read directly from the deep resistivity curve. Corrections are not necessary for wellsite calculations, or most other log analysis work.

By convention, the deep resistivity curve is coded as a dashed line, simply remember deep = dashed.

When deep, medium, and shallow or micro resistivity curves are all available on the same log, the shallowest device is coded as a solid line, the intermediate device as a light dashed line, and and the deep device as a heavy dashed line. Figure 4-1 portrays this situation.

Table 4-1 classifies the resistivity tools by depth of

Major Assumptions for Rt Determination:
1. The proper Rt tool was run.

Major Cautions in Rt Determination:
Check to see that the proper Rt tool was run, see Chart 4-1 below.

If the wrong Rt tool was run, use the appropriate charts in the service company manual to correct the Rt value, or at least get an idea of the size of the error in the Rt value. Correction charts exist for all the various types of resistivity tools that every existed, however only the recent and most common charts are in the modern day chartbooks.

Invasion may prevent the deep resistivity value from measuring Rt directly. Provided there are at least three depths of investigation then invasion corrections can be made, (again from chart books). In conclusion, quick look computations can usually be made using Rt taken directly from the Deep resistivity measurements, however caution may be necessary.

There has been much development in the area of resistivity measurements in recent years. The implementation of multiple array technology and computer software and hardware have added both complexity and accuracy to the measurement of Rt. This technology is available to tackle the more difficult reservoir analysis as well as the more mundane. Software solutions have also been developed that more accurately describe the formation model. These solutions can be even applied to older array resistivity data with improved results.
Resistivity inversion software is now available. This technique can be used to model the expected response of various resistivity tools along a proposed drill path.

Section 5. Porosity Derivation

Porosity values are most commonly derived from the density, neutron, or sonic alone, or from the neutron and density in combination. We can also computer porosity from resistivity measurements (a variation of the previous chapters discussions). Modern nuclear magnetic Resonance tools are available which can predict the formation fluid types as well as volume. This chapter is devoted to the basic calculations and derivations of porosity from the neutron, density and acoustic measurements.

Use of any single porosity device requires assumptions for matrix values and fluid values. The neutron-density combination largely eliminates these assumptions and is recommended for that reason.

5.1 Density Porosity