CHAPTER SEVEN: CALCULATING POROSITY

Table Of Contents
7.00 Introduction To This Chapter
7.01 Definitions Of Porosity
7.02 Visual Indications Of Porosity
7.03 Scaling Logs In Porosity Units
7.04 Porosity From Compressional or Shear Sonic Log
7.05 Porosity From The Density Log
7.06 Porosity From Density Porosity Log With Matrix Offset
7.07 Porosity From Old Style Neutron Logs
7.08 Matrix Offset For Neutron Logs
7.09 Porosity From The Neutron Log
7.10 Summary Of One Log Porosity Methods
7.11 Quick Methods For Density Neutron Crossplot Calculations
7.12 Shaly Sand Crossplot (Density Neutron) With Matrix Offset
7.13 Complex Lithology Crossplot (Density Neutron)
7.14 Bulk Volume Water Crossplot (Dual Water Model)
7.15 Sonic Neutron Crossplot
7.16 Sonic Density Crossplot
7.17 Summary Of Crossplot Methods
7.18 Discussion Of Gas Correction Methods <<< New
7.19 Porosity From Microlog
7.20 Porosity From Shallow Resistivity Logs
7.21 Porosity From Deep Or Medium Resistivity Log
7.22 Summary Of Porosity From Resistivity Methods
7.23 Non-Porous Lithology Triggers
7.24 Material Balance For Porosity (Maximum Porosity)
7.25 Useful Porosity
7.26 Porosity from Nuclear Magnetic Log
7.27 Fracture Porosity
7.28 Porosity from Three-Mineral Lithology
7.29 Selection Of Porosity Method
7.30 Effective Porosity Routines
7.31 Calibrating Porosity to Core and Sample Data
7.32 Sensitivity Analysis
7.33 In Conclusion
7.34 Exercises For Chapter Seven
7.35 Bibliography For Chapter Seven

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Publication History: Originally published as Chapter Seven of the Log Analysis Handbook, Pennwell 1986. Sections 7.25 through 7.32 added for this electronic edition Feb 2001. Shear sonic porosity added to Section 7.04 Aug 2003. Simplified complex lithology model added to Section 7.11 Oct 2003. Section 7.00 revised Sept 2001 and Oct 2003.

CHAPTER SEVEN: CALCULATING POROSITY

7.00 Introduction to This Chapter
The next step in quantitative analysis, after finding shale volume, is to estimate porosity.

There are numerous porosity indicating logs, as shown in the box at left, and many flavours of each, depending on the age, design, and logging environment. Generic analysis equations, based on the Log Response Equation, for each basic tool type are contained in this Chapter. They will work for almost all available tool types. There may be rare occasions when a customized analysis model might be required.

All the porosity models require some assumptions about such things as fluid type and matrix rock properties. With the exception of the resistivity log formula, used for analysis of ancient logs, the methods involve corrections for the effects of shale.

FIGURE 7.00: The Effect of Shale on Porosity

Shale corrections are applied to porosity logs to determine effective porosity, as shown in the illustration above. Since shale contains some water, this water must be subtracted from the total porosity as measured by conventional logging tools. The mathematical method for finding shale volume is the same for all the shale distribution types, but the method for applying the shale correction to the porosity varies. This Chapter deals with clean sands or shaly sands dispersed or structural shale. Chapter Seventeen covers laminated shaly sands.

Correcting for shale is only half the battle. The other half is to correct for the mineral composition of the rocks. In most carbonate reservoirs, the lithology is usually reasonably well known from sample descriptions or can be determined from log response, so this step is relatively straightforward.

This is not true in sandstones because the mineral makeup of the sand is NOT usually described in much detail. There is a universal trend to give sandstones the physical properties of pure quartz, but this is almost universally NOT appropriate. Most sandstones contain other minerals such as mica, volcanic rock fragments, calcite, dolomite, anhydrite, and ferrous minerals, as well as the shale and clay described above. All of these minerals have different density, acoustic, and neutron properties than quartz. If a sandstone is assumed to be pure quartz when it is not, the commonly used properties of quartz will provide a pessimistic porosity answers.

Thus, authors and service company manuals that present mineral properties for “sandstone” are misleading their audience into believing these properties are constant. In more than 40 years of petrophysical analysis, I have never seen a thin section or XRD report that gave an assay of 100% quartz in any petroleum reservoir. A 100% quartz sand is very rare. If anyone doubts this statement, look at the PEF curve. If it reads more than 1.8, you have “quartz plus other things” in your sandstone.

There is a story (it may even be true) that reserves for the early North Sea discoveries were seriously underestimated because the mica in the sands was not accounted for properly. The engineers used density log porosity without correcting for the real matrix density. If true, good engineering practice would have undersized all the offshore equipment and early cash flow and rate of return on investment would have been significantly reduced. If the myth that sandstone is pure quartz is perpetuated, there will be more economic blunders of this type.

To further confuse the uninitiated, many logs show data on a "porosity" scale. These log curves are transforms of some measured physical property to an approximate porosity based on some arbitrary parameters. Examples are density, neutron, or sonic porosity on so-called Sandstone, Limestone, or Dolomite porosity scales. Porosity as defined by these transforms is only directly useful if there is no shale, the scale matches the rock mineralogy. and there are no accessory minerals. Real reservoirs are rarely this simple. DO NOT use these porosity transforms without further analysis unless all the arbitrary assumptions used to create them match exactly the rock you are analyzing.

Some people call these porosity curves an “interpretation”. They are not. They are merely a transform of the raw data to a more attractive scale. The difference between a transform and an interpretation is critical. Interpretation infers some intelligent thought went into creating and understanding the result. The service company running the log does not provide interpretations. YOU are the interpreter,

There are endless cases where a transform to an inappropriate porosity scale has caused millions in losses due to poorly informed analysts who see “gas cross over” when there is no gas, or who read porosity directly from the transform and either seriously over estimate or under estimate reservoir effective porosity.

In spite of these comments, a number of charts and tables in this Chapter and elsewhere in this Handbook show the word "sandstone' when they really should say "quartz". I have not edited the charts and tables taken from common sources, such as service company chart books, so the common usage of incorrect terminology is repeated even here.

7.01 Definitions of Porosity
Porosity is the volume of the non-solid portion of the rock filled with fluids, divided by the total volume of the rock.

Primary porosity is the porosity developed by the original sedimentation process by which the rock was created. In reports, it is often referred to in terms of percentages, while in calculations it is always a fraction.

To acquire an appreciation for the values of porosity generally encountered, assume round balls of the same size are stacked on top of each other in columns. Calculations will show a porosity of 47.6%. Spherical sand grains 1/10 the size of the balls stacked one on top of the other will have the same porosity, 47.6%.

If the same balls are packed in the closest possible arrangement in which the upper ball sits in the valley between the four lower balls, each touching, the porosity is reduced to 25.9%. Again, changing the size of the balls will not change the porosity as long as all the balls are the same size. Mixing the sizes of the balls will create lower porosity, since small ones can fit in spaces created between the larger ones.

The highest porosity normally anticipated is 47.6%. A more probable porosity is in the mid-twenties. The normal range of porosities in granular systems is 5% to 35%.

In general, porosities tend to be lower in deeper and older rocks. This decrease in porosity is primarily due to overburden pressure stresses on the rock, and cementation. There are many exceptions to this general trend, when normal overburden conditions do not prevail.

Shales closely follow the same porosity depth trend as sandstones, except that porosities are normally lower in shales. For example, in a recent mud the porosity may measure about 40%. It decreases rapidly with depth and overburden pressure until, at a depth of about 10,000 feet, normal porosities are less than 5%. Shales are plastic and therefore, compress more easily than sands.
These basic trends of porosity versus depth are not as noticeable in carbonates, where porosity is more a function of depositional environment and secondary processes, both unrelated to depth of burial.

Porosity in a real shale is not effective; that is, the water cannot move as quickly as in a sandstone with the same apparent porosity. Water in shale can be expelled over large geologic time periods, but it will not flow in the usual sense of the word.

However, many intervals that have been traditionally thought of as "shale" are really silty shales or sandy shales. These may have sufficient porosity to store hydrocarbons that might flow. This is especially true for gas, and many "gas shales" are silty shales with effective porosity. Other gas shales are mostly shale and gas is stored on the surface of fractures within the shale. This is adsorbed gas.

Secondary porosity is created by processes other than primary cementation and compaction of the sediments. An example of secondary porosity can be found in the solution of limestone or dolomite by ground waters, a process which creates vugs or caverns. Fracturing also creates secondary porosity. Dolomitisation results in the shrinking of solid rock volume as the material transforms from limestone to dolomite, giving a corresponding increase in porosity.

In most cases, secondary porosity results in much higher permeability than primary granular porosity.

The use of the term, Secondary Porosity Index (SPI), by log analysts has led to much confusion. The term means the porosity defined by the difference between porosity derived from the sonic log and the primary porosity. The primary porosity is usually defined by core analysis or the density neutron log. Depending on the shape and size of the vugs, fractures, or caverns, the SPI may or may not be a good indication of secondary porosity.

The above discussion covers the geological definitions of porosity. Petrophysicists, log analysts, and engineers use more specific terms based on the log analysis model described in Chapter Four. Here are the definitions that derive from that model.

Non-useful porosity occurs as tiny pores that do not connect to any other pores. They are almost invariably filled with immoveable water and do not contribute to useful reservoir volume or energy. Such pores occur in silt, volcanic rock fragments in sandstones, and in micritic, vuggy, or skeletal carbonates. The NMR may see some of this non-useful porosity; the jury is still out.

Porosity derived directly from a log without correction for shale content, is termed apparent or total porosity. If the zone has no shale, the total porosity equals the effective porosity. Should the zone contain shale, corrections must be applied to obtain effective porosity. DO NOT USE LOG READINGS DIRECTLY UNLESS THERE IS ZERO SHALE CONTENT.

This warning also applies to logs recorded in porosity units when the log scale does not match the actual lithology. For example, a density, neutron, or sonic log can be run on sandstone, limestone, or dolomite scales. While these scales have many valuable uses, they will give erroneous results unless the rock mineralogy exactly matches the scale definition. A log recorded on a limestone scale in a clean sandstone, shaly sandstone, or dolomite needs further data processing before it will give the correct answer.

Various methods are presented here to calculate porosity from individual or combinations of two or more logs. Two log combinations are termed crossplot methods, since the log data can be plotted on the X and Y axes of a graph.

Three or more log combinations require solution by simultaneous equations, and are usually done on a computer.


7.02 Porosity Overlay/Porosity Playback Log
All porosity logs have been recorded in such a fashion as to deflect to the left when porosity increases. This also occurs in shale zones, which creates a conflict when attempting to do a visual log interpretation since both shale content and porosity increase to the left. Use the GR and SP to discriminate shale from porous rock.

FIGURE 7.01: Porosity Playback Log for Classic Example 1

Figure 7.0l illustrates the three usual porosity logs on compatible scales, displaying that the deflection for increased porosity is always to the left.

Porosity derived from the resistivity log is also shown. This presentation is called a porosity playback log and is created in the computer, but can usually be produced by overlaying or tracing logs on compatible scales.

Non-compatible scales may be used but additional care is required.

When porosity from the resistivity log tracks the porosity curves, then the zone is probably water bearing or shaly. If the porosity from the resistivity log departs to the right of the porosity curves, the zone probably contains hydrocarbon, is tight (has no porosity) or is coal. The departure to the right was dubbed the "Mae West Effect" many years ago, but it is no longer fashionable to use this term.


7.03 Scaling Logs in Porosity Units
If logs are not recorded on porosity scales, or scales are inappropriate, it is convenient to label the required porosity scale on the log.

Table 7.0l, illustrates approximate porosity scales for a number of individual logs. These values should be memorized so that the analyst can derive approximate porosity at any time without reference to chartbooks or calculators. Porosity obtained in this manner will presumably be too high as no shale correction has been made. A mental deduction for the amount of shale, estimated from the gamma ray or SP log, should be included prior to finalizing any visual interpretation.

TABLE 7.01: Scaling Porosity Logs

Density neutron logs can be displayed on sandstone scales or limestone scales, regardless of rock type. This is a function of a switch setting in the logging truck, which allows a sandstone scale to be run in limestone rocks and vice-versa. If the scale name (e.g. sandstone) does not coincide with the rock type (e.g. limestone), the rules in Table 7.0l should be applied to derive the appropriate scale. When using charts or calculators as opposed to visual methods, use the rules pertaining to those methods, and not Table 7.0l.

TABLE 7.02: Scaling Porosity Logs

Porosity found by scaling the log in porosity units is termed the Total Porosity (PHIt), and will vary for each log, AND IS NOT THE FINAL ANSWER. NOTE: GAS AND SHALE AFFECT THE APPARENT POROSITY, SO POROSITY DETERMINED BY SCALING THE LOG IS MERELY THE FIRST STEP IN A VISUAL INTERPRETATION

The qualitative response of basic porosity logs to gas and shale are given in Table 7.02. Use these rules to modify your opinion of porosity determined by scaling the logs, or follow through with detailed hydrocarbon and shale corrections described later in this chapter.

To apply the rules in Table 7.0l, draw the scale on the log using the zero and 0.1 points listed. Label the 0.2, 0.3, 0.4 and 0.5 points by shifting an equal distance for each additional 0.1 fraction of porosity.

For example, on English units sonic logs, to create a sandstone porosity scale, mark the 0.0 porosity point at 55.5 usec/ft and the 0.1 point at 68.5 usec/ft. Add another 13 usec/ft for each 0.1 extra porosity to find the 0.2 and 0.3 porosity points. See Figure 7.02.

FIGURE 7.02: Scaling Porosity Logs

As a second example, assume a limestone unit neutron porosity scale, and convert it to a sandstone unit scale by exercising the rule "add 0.04" to get sandstone from limestone units.

The third example is the case of scaling an obsolete neutron log recorded in counts per second or other arbitrary units. The usual approach is to pick a low point on the scale, and label it as 0.25 or 0.30 porosity units. Then label a high scale point as 0.01 or 0.02 porosity units and scale logarithmically between these two points.

These three examples can be found in Figure 7.02.

The data for the sonic log for Classic Example 1 is shown with its appropriate porosity scale in the correct units for further work, as shown in Figure 7.03.

FIGURE 7.03: Scaling the Sonic Log for Classic Example 1

The density neutron log for this example is already on a porosity scale in the correct sandstone porosity units, as shown in Figure 7.01.


7.04 Porosity from the Sonic Log
The response equation for the sonic log follows the classical form:
DELT = PHIe * Sxo * DELTw (water term)
+ PHIe * (1 - Sxo) * DELTh (hydrocarbon term)
+ Vsh * DELTsh (shale term)
+ (1 - Vsh - PHIe) * Sum (Vi * DELTi) (matrix term)

WHERE
DELTh = log reading in 100% hydrocarbon
DELTi = log reading in 100% of the ith component of matrix rock
DELT = log reading
DELTsh = log reading in 100% shale
DELTw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)

To solve for porosity from the sonic log, we assume DELTh, DELTi, DELTsh, DELTw, and Vsh are known. We also assume DELTw = DELTh and Sxo = 1.0 when no gas is present. If gas is indicated, we make assumptions about DELTh and Sxo, usually in the form of a correction factor to the gas free case. The usual result is:

PHIsc = (DELT - (1 - Vsh) * DELTMA - Vsh * DELTSH) / (DELTW - DELTMA)

The response equation is not rigorous and many exceptions are noted below.

The rules for sonic logs in Table 7.01 and 7.02, which represent simplified cases of the response equation, can be converted to calculator or computer use by the following equations. This porosity method is one of the most common calculations on older wells. The shale correction is very important and should not be ignored.

Calculate sonic compaction correction
1: CP = max (1, CDTSH / (100 + 228 * (IF DEPTHUNIT$ = "METRIC")))

Calculate total sonic porosity
2: PHIS = (DELT - DELTMA) / (DELTW - DELTMA) / CP

Correct sonic porosity for shale
3: PHISSH = (DELTSH - DELTMA) / (DELTW - DELTMA) / CP
4: PHIsc = PHIS - Vsh * PHISSH

Correct sonic porosity for gas effect
5: IF SONICGASSWITCH$ = "ON"
6: THEN PHIsc = KS * PHIS

WHERE:
CDTSH = shale travel time for compaction correction (usec/ft or usec/m)
CP = compaction factor (fractional)
DELT = sonic log reading in zone of interest (usec/ft or usec/m)
DELTMA = sonic log reading in l00% matrix rock (usec/ft or usec/m)
DELTSH = sonic log reading in l00% shale (usec/ft or usec/m)
DELTW = sonic log reading in 100% water (usec/ft or usec/m)
KS = sonic log gas correction factor
PHIS = porosity from sonic log (corrected for compaction if needed) (fractional)
PHIsc = porosity from sonic log by Wyllie method (fractional)
PHISSH = apparent sonic porosity of 100% shale after compaction correction (if needed) (fractional)
Vsh = shale volume (fractional)

COMMENTS:
Of the three "one-log" porosity methods, the sonic corrected for shale is the preferred one for wells drilled after 1957 and before 1965. However, crossplot methods or the density log corrected for shale are usually better if the log data is available.

The graphical solution for these formulae is provided in Figure 7.04. Simpler charts exist which do not include the shale or fluid correction. If any significant amount of shale exists, do not use simple charts.

FIGURE 7.04: Chart for Estimating Shale Corrected Sonic Porosity

Use the compaction correction only if CDTSH > 100 (for English units) or CDTSH > 328 (for Metric units). In western North America, this is normally required when above 3,000 - 4,000 feet (900 - l,200m).

KS is in the range 0.7 to 1.0 depending on gas density invasion and local experience. It can be derived by comparing the calculated porosity with the true porosity from cores or density neutron crossplot methods.

Use gas correction only if PHIS is too high compared to other sources, only if the zone is clean and does not need shale corrections, and if gas is known to be present. The need for this correction is rare. It is very unlikely that a gas correction will be needed in shaly sands since invasion should be relatively deep.

Another way of making gas corrections in both methods is to change DELTW to a higher value, representing the travel time of sound in a mixture of gas and water. This value depends on water saturation in the invaded zone, pressure, temperature and gas compressibility. Values in the range of 600 usec/ft (1900 usec/m) at shallow depths to 300 usec/ft (950 usec/m) at 6000 feet (2000 meters) are recommended as a starting point.

If log is in porosity units, skip Step 1 and Step 5, and read PHIS and PHISSH directly from the log. If porosity scale is in sandstone units and rock type is limestone (or vice versa), make appropriate adjustments as per Table 7.01.

CDTSH may be higher if depth is less than 3,000 ft (1,000m). Usually set CDTSH equal to DELTSH, with a minimum of 100 (English) or 328 (Metric).

CDTSH can be calculated if true porosity of a clean zone is known from core, neutron, or density log data:

CDTSH = PHIS / PHItrue * DELTSH
OR: CP = PHIS / PHItrue

WHERE:
CDTSH = shale travel time for compaction correction (usec/ft or usec/m)
CP = compaction factor (fractional) (usec/ft or usec/m)
DELTMA = sonic log reading in 100% rock matrix (usec/ft or usec/m)
DELTSH = sonic log reading in 100% shale (usec/ft or usec/m)
DELTW = sonic log reading in 100% water (usec/ft or usec/m)
PHIS = sonic log porosity in clean sand (fractional)
PHItrue = actual porosity in clean sand from core or density data (fractional)

The Hunt-Raymer method is a newer formula which is a non-linear calibration of observed porosity versus log response data. It should be used in clean sands and carbonates only, or log data may be corrected for shale first. It can be used in un-compacted sands without the compaction correction described in the Wyllie method given above. The algorithm is derived from the following empirical relationship: VELOG = VELMA * ((1 - PHIe) ^ 2) + VELW * PHIe

This can be solved for porosity in the following way:

Calculate sonic log reading corrected for shale:
1: DELTc = DELT - Vsh * (DELTSH - DELTMA)

Calculate sonic porosity
2: C = DELTMA / (2 * DELTW)
3: PHIShr = 1 - C - (C ^ 2 - DELTMA / DELTW + DELTMA / DELTc) ^ 0.5

WHERE:
C = intermediate term
DELT = sonic log reading in zone of interest (usec/ft or usec/m)
DELTc = sonic log reading corrected for shale (usec/ft or usec/m)
DELTMA = sonic log reading in l00% matrix rock (usec/ft or usec/m)
DELTSH = sonic log reading in l00% shale (usec/ft or usec/m)
DELTW = sonic log reading in 100% water (usec/ft or usec/m)
` PHIShr = porosity from sonic log by Hunt-Raymer method (fractional)
VELOG = sonic velocity log reading (ft/sec or m/sec)
VELMA = sonic velocity log reading in 100% matrix (ft/sec or m/sec)
VELW = sonic velocity log reading in 100% water (ft/sec or m/sec)
Vsh = shale volume (fractional)

COMMENTS:
A graphical solution for the Hunt-Raymer method, with no shale correction, is given in Figure 7.04A.

FIGURE 7.04A: Sonic Log Porosity from Hunt-Raymer Method (curved lines) and Wyllie Method (straight lines) - No Shale Corrections

Although the original paper does not discuss shale corrections, they are essential. Gas corrections similar to those used in the Wyllie method can be used if needed. The answer porosity will be too high in gas if the corrections are not made. The method is not universally applicable and should be tested in each area before use.

Another way of making gas corrections in both methods is to change DELTW to a higher value, representing the travel time of sound in a mixture of gas and water. This value depends on water saturation in the invaded zone, pressure, temperature and gas compressibility. Values in the range of 600 usec/ft (1900 usec/m) at shallow depths to 300 usec/ft (950 usec/m) at 6000 feet (2000 meters) are recommended as a starting point.

RECOMMENDED PARAMETERS:
See Wyllie method discussed above

NUMERICAL EXAMPLE:
1. Wyllie Method - data from Sand "D" of Classic Example 1.
DELT = 300 usec/m
DELTSH = 328 usec/m
CDTSH = 328 usec/m
DELTMA = 182 usec/m
DELTW = 616 usec/m
Vsh = 0.33

CP = 328 / 328 = 1.0
Therefore compaction correction is not needed.

PHIS = (300 - 182) / (616 - 182) / 1.0 = 0.27
PHISSH = (328 - 182) / (616 - 182) / 1.0 = 0.34
PHIsc = 0.27 - 0.33 * 0.34 = 0.16
PHIsc is not too high, and no gas is known to be present. Hence, no gas correction is made.

2. Hunt-Raymer Method - data from Sand D above.
DELTc = 300 - 0.33 * (328 - 182) = 251 usec/m
C = 182 / (2 * 616) = 0.147
PHIShr = 1 - 0.147 - (0.147 ^ 2 - 182 / 616 + 182 / 251) ^ 0.5 = 0.18

3. Wyllie Method - data from Sand "C"
DELT = 380 usec/m
DELTSH = 328 usec/m
CDTSH = 328 usec/m
DELTMA = 182 usec/m
DELTW = 616 usec/m

Vsh = 0.0
CP = 328 / 328 = 1.0
PHIS = (380 - 182) / (616 - 182) / 1.0 = 0.46
PHISSH = (328 - 182) / (616 - 182) = 0.36
PHIsc = 0.46 - 0.0 * 0.36 = 0.46
PHIsc is too high due to gas effect - assume KS = 0.75
PHIsc = 0.75 * 0.46 = 0.33

4. Hunt-Raymer Method - data from Sand C above.
DELTc = 380 - 0.00 * (328 - 182) = 380 usec/m
C = 182 / (2 * 616) = 0.147
PHIShr = 1 - 0.147 - (0.147 ^ 2 - 182 / 616 + 182 / 380) ^ 0.5 = 0.40
Porosity is too high due to gas effect - assume KS = 0.80.
PHIsc = 0.80 * 0.40 = 0.32

5. Wyllie Method - data from Sand "A"
DELT = 375 usec/m
DELTSH = 460 usec/m
CDTSH = 460 usec/m
DELTMA = 182 usec/m
DELTW = 616 usec/m

Vsh = 0.0
CP = 460 / 328 = 1.40
PHIsc = PHIS = (375 - 182) / (616 - 182) / 1.40 = 0.31
No gas correction is required.
No shale correction is required.

6. Hunt-Raymer Method - data from Sand A above.
DELTc = 375 - 0.33 * (460 - 182) = 375 usec/m
C = 182 / (2 * 616) = 0.147
PHIShr = 1 - 0.147 - (0.147 ^ 2 - 182 / 616 + 182 / 375) ^ 0.5 = 0.39

This result is a little high compared to the more conventional method.

The newer sonic logs record shear travel time as well as the compressional travel tine. The compressional data is processed as discussed above under the Wyllie and Raymer-Hunt methods. Shear travel time can be used in the Wyllie equation, using fictitious values for fluid travel time. There is very little fluid effect on shear data so there is no gas correction.

Calculate total sonic porosity
1: PHIshear1 = (DTS - DTMA_S) / (DTW_S - DTMA_S)

Correct sonic porosity for shale
2: PHISSH_S = (DTSH_S - DTMA_S) / (DTW_S - DTMA_S)
3: PHIshear = PHIshear1 - Vsh * PHISSH_S

WHERE:
DTS = shear sonic log reading in zone of interest (usec/ft or usec/m)
DTMA_S = shear sonic log reading in l00% matrix rock (usec/ft or usec/m)
DTSH_S = shear sonic log reading in l00% shale (usec/ft or usec/m)
DTW_S = (fictitious) shear sonic log reading in 100% water (usec/ft or usec/m)
PHIshear1 = porosity from shear sonic log before shale correction (fractional)
PHIshear = porosity from shear sonic log by Wyllie method (fractional)
PHISSH_S = apparent shear sonic porosity of 100% shale (fractional)
Vsh = shale volume (fractional)

COMMENTS:
Shear travel time is more sensitive to porosity than compressional data.

No gas correction is needed.

The measurement can usually be made through casing so this is a good choice for cased hole logging.

There is no record of a compaction correction being applied, but this may be needed. Comparison to core porosity or density neutron crossplot porosity will indicate when such a correction is needed.

7.05 Porosity from the Density Log
The response equation for the density log in porosity units follows the classical form:
PHID = PHIe * Sxo * PHIDw (water term)
+ PHIe * (1 - Sxo) * PHIDh (hydrocarbon term)
+ Vsh * PHIDsh (shale term)
+ (1 - Vsh - PHIe) * Sum (Vi * PHIDi) (matrix term)

WHERE:
PHIDh = log reading in 100% hydrocarbon
PHIDi = log reading in 100% of the ith component of matrix rock
PHID = log reading
PHIDsh = log reading in 100% shale PHIDw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)

To solve for porosity from the density log, we assume PHIDh, PHIDi, PHIDsh, PHIDw, and Vsh known. We also assume PHIDw = PHIDh and Sxo = 1.0 when no gas is present. If gas is indicated, we make assumptions about PHIDh and Sxo, usually in the form of a correction factor to the gas free case, as described later.

Since PHIDi = 0 and PHIDw = 1.0, the usual result is:
PHIdc = PHID - Vsh * PHIDSH

This response equation is rigorous.

The rules for density logs in Tables 7.01 and 7.02, based on the response equation, are translated algebraically by the following formulae:

Calculate density porosity from density data.
1: PHID = (DENS - DENSMA) / (DENSW - DENSMA)

Apply density shale correction:
2: PHIDSH = (DENSH - DENSMA) / (DENSW - DENSMA)
3: PHIdc = PHID - Vsh * PHIDSH

Apply density gas correction.
4: IF DENSITYGASSWITCH$ = "ON"
5: THEN PHIdc = KD * PHIdc

WHERE:
DENS = density log reading in zone of interest (gm/cc or Kg/m3)
DENSMA = density log reading in 100% matrix rock (gm/cc or Kg/m3)
DENSSH = density log reading in 100% shale (gm/cc or Kg/m3)
DENSW = density log reading in 100% water (gm/cc or Kg/m3)
KD = density log gas correction (fractional)
PHID = porosity from uncorrected density log (fractional)
PHIdc = porosity from density log corrected for shale (fractional)
PHIDSH = apparent density log porosity of 100% shale (fractional)
Vsh = shale volume (fractional)

COMMENTS:
A graphical solution, with shale correction, is in Figure 7.05.

FIGURE 7.05: Chart for Estimating Shale Corrected Density Porosity

The density log corrected for shale is a very good approximation to porosity, but the log was not common before 1965, so sonic or neutron methods may be necessary for wells drilled before that time.

KD is in the range of 0.5 - 1.0 depending on invasion, gas density and local experience. A correction is almost always needed if gas is present.

Use gas correction only if PHIdc is too high compared to other sources and if gas is known to be present. This correction may be necessary even in shaly sands, since the depth of investigation of the density log is deep enough to see beyond the flushed zone.

If log is in porosity units, use rules in Table 7.01 to get appropriate porosity scale for the lithology being encountered or see next section. Also disregard Step 1 and Step 4, and read PHID and PHIDSH directly from the log.

If density porosity data is in percent, rather than fractional, divide the data values by 100 before Step 2 and 3 are applied.

No compaction correction is made to density log data.

RECOMMENDED PARAMETERS:
See Section 7.06.

NUMERICAL EXAMPLE:
1. Assume a zone with:
DENS = 2.15 gm/cc
DENSW = 1.00 gm/cc
DENSMA = 2.65 gm/cc
Vsh = 0.33
DENSSH = 2.60 gm/cc
PHID = (2.15 - 2.65) / (1.00 - 2.65) = 0.30
PHIDSH = (2.60 - 2.65) / (1.00 - 2.65) = 0.03
PHIdc = 0.30 - 0.33 * 0.03 = 0.29
No gas correction is required.

7.06 Porosity From Density Porosity Log With Matrix Offset
One step that is often required is to convert apparent porosity on the density log into density units, then reconstitute porosity from this value corrected for a desired matrix and fluid value. This is done by rearranging the response equation of the previous section.

The response equation for the density log in density units follows the usual form:
DENS = PHIe * Sxo * DENSw (water term)
+ PHIe * (1 - Sxo) * DENSh (hydrocarbon term)
+ Vsh * DENSsh (shale term)
+ (1 - Vsh - PHIe) * Sum (Vi * DENSi) (matrix term)

WHERE:
DENSh = log reading in 100% hydrocarbon
DENSi = log reading in 100% of the ith component of matrix rock
DENS = log reading
DENSsh = log reading in 100% shale
DENSw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)

To solve for porosity from the density log, we assume DENSh, DENSi, DENSsh, DENSw, and Vsh are known. We also assume DENSw = DENSh and Sxo = 1.0 when no gas is present. If gas is indicated, we make assumptions about DENSh and Sxo, usually in the form of a correction factor to the gas free case, as described later.

Calculate density from density porosity.
1: DENSm = (PHID * 1.00 + (1 - PHID) * (2.65 + 0.06 * (IF LOGUNIT$ = “LIMESTONE")))
* (1 + 999 * (IF DEPTHUNIT$ = "METRIC"))

Calculate shale density.
2: DENSSHm = (PHIDSH * 1.00 + (1 - PHIDSH) * (2.65 + 0.06 * (IF LOGUNIT$ =
"LIMESTONE"))) * (1 + 999 * (IF DEPTHUNIT$ = "METRIC"))

Calculate porosity with new matrix and fluid.
3: PHIDm = (DENSm - DENSMA) / (DENSW - DENSMA)
4: PHIDSHm = (DENSSHm - DENSMA) / (DENSW - DENSMA)
5: PHIdc = PHIDm - Vsh * PHIDSHm

Apply density gas correction.
6: IF DENSITYGASSWITCH$ = "ON"
7: THEN PHIdc = KD * PHIdc

WHERE:
DENSSHm = density log reading in 100% shale reconstituted from density porosity data (gm/cc or Kg/m3)
DENSm = density value reconstituted from density porosity data (gm/cc or Kg/m3)
DENSMA = matrix density (gm/cc or Kg/m3)
DENSW = fluid density (gm/cc or Kg/m3)
PHID = porosity from uncorrected density log (fractional)
PHIdc = porosity from density log corrected for shale (fractional)
PHIDm = density porosity log reading corrected for matrix offset (fractional)
PHIDSH = density porosity log reading in 100% shale (fractional)
PHIDSHm = density porosity log reading in 100% shale corrected for matrix offset (fractional)
Vsh = volume of shale (fractional)

COMMENTS:
The graphical solution to these formulae is provided in Figure 7.05, shown in the previous section. As for the sonic log, simpler charts exist. However they should not be used if shale is present. All comments from Section 7.05 also apply.

WHERE:
DENSMA = matrix density (gm/cc or Kg/m3)
DENSW = fluid density (gm/cc or Kg/m3)
DENSSH = shale density (gm/cc or Kg/m3)

NUMERICAL EXAMPLE:
1. Data from Sand "D" in Classic Example 1
PHID = 0.12
PHIDSH = 0.03
Vsh = 0.33
Data is already in porosity units, so conversion to porosity units is not required.
No gas is known and log reading is not too high, so no gas correction is needed.
PHIdc = 0.12 - 0.33 * 0.03 = 0.11

2. Data from Sand "C" in Classic Example 1
PHID = 0.33
PHIDSH = 0.30
Vsh = 0.0
Log is already in porosity units, but porosity is too high due to gas.
PHIdc = 0.9 * 0.33 = 0.30
No shale correction is necessary.

3. Convert data to equivalent dolomite porosity with no change in fluid properties.
PHID = 0.30 (on sandstone scale)
DENSW = 1.00 gm/cc
DENSMA = 2.83 gm/cc (output units)
PHIDSH = 0.03
Vsh = 0.0
DENS = 0.30 * 1.00 + (1 - 0.30) * 2.65 = 2.15 gm/cc
DENSSH = 0.03 * 1.00 + (1 - 0.,03) * 2.65 = 2.60 gm/cc
PHIDm = (2.15 - 2.83) / (1.00 - 2.83) = 0.37
PHIDSHm = (2.60 - 2.83) / (1.00 - 2.83) = 0.12
PHIdc = 0.37 - 0.0 * 0.12 = 0.37

This value is quite high for a dolomite. Therefore, a gas correction should be considered, or else the rock is not a dolomite after all.

7.07 Porosity from Old Style Neutron Logs
For old style GRN or un-scaled neutron logs recorded in counts per second or API units, a porosity scale must be derived by the analyst. A logarithmic scale can be applied algebraically with the following formulae using the high porosity/low porosity method.

1: SLOPE = (log (PHIHI / PHILO)) / (CPSHI - CPSLO)
2: INTCPT = PHIHI / 10 ^ (CPSHI * SLOPE)
3: PHIn = INTCPT * 10 ^ (SLOPE * NCPS)

WHERE:
CPSHI = GRN counts at high porosity point (cps)
CPSLO = GRN counts at low porosity point (cps)
NCPS = neutron log reading in CPS or arbitrary units (cps)
PHIHI = high porosity point (fractional)
PHILO = low porosity point (fractional)
PHIn = apparent neutron log porosity, uncorrected for shale (fractional)

COMMENTS:
The graphical solution to this formula is given in Figure 7.06. Complete gas, shale and matrix corrections will still be required and are detailed in the following sections.

FIGURE 7.06: Chart for Estimating Porosity from Neutron Counts per Second - no shale correction

A large number of charts for specific tools, spacings, borehole conditions and rock types are available from service companies.

RECOMMENDED PARAMETERS:
PHIHI should be in the range 0.20 to 0.35.
PHILO should be in the range 0.01 to 0.05, and cannot be zero.

NUMERICAL EXAMPLE:
1. Assume an old GRN log where:
PHIHI = 0.30
PHILO = 0.01
NCPS = 2500 cps
CPSHI = 1500
CPSLO = 4500
SLOPE = (log (0.30 / 0.01)) / (1500 - 4500) = - 0.000492 (rounded to - 0.0005)
INTCPT = 0.30 / 10 ^ (1500 * (-0.0005)) = 1.6432
PHIn = 1.6432 * 10 ^ (-.0005 * 2500) = 0.096

7.08 Matrix Offset for Neutron Logs
It is often necessary to rescale a neutron log, which is already in porosity units, for lithology.

Sandstone porosity units to limestone units.
CASE 1: PHINm = PHIN - 3 - 1 * (IF NEUTRONTYPE$ = "CNL")
Limestone porosity units to sandstone units.
CASE 2: PHINm = PHIN + 3 + 1 * (IF NEUTRONTYPE$ = "CNL"
Mud cake thickness correction (SNP only).
CASE 3: PHINm = PHIN - 0.01 * max (0, CAL - BITZ) / (1 + 24.4 (IF DEPTHUNIT$ = "METRIC"))

If the log is recorded in limestone units or has been shifted to approximate limestone units, and a correction for more accurate lithology is desired, use the following formulae:

If lithology is sandstone and tool type is SNP.
CASE 4: PHINm = 0.024 + 1.021 * (PHIN ^ (-22.2 * PHIN - 1.96))
If lithology is dolomite and tool type is SNP.
CASE 5: PHINm = - 0.00434 + 0.749 * PHIN + 0.60 * (PHIN ^ 2)
If lithology is sandstone and tool type is CNL.
CASE 6: PHINm = 0.039 + 1.021 * (PHIN ^ (-22.2 * PHIN - 1.96))
If lithology is dolomite and tool type is CNL.
CASE 7: PHINm = -0.01259 + 0.389 * PHIN + 1.4 * (PHIN ^ 2)
If no lithology correction is needed.
CASE 8: PHINm = PHIN

WHERE:
BITZ = bit size (inches or mm)
CAL = caliper (inches or mm)
PHIN = original neutron log reading
PHINm = apparent neutron log porosity corrected for lithology (fractional)

COMMENTS:
These lithology adjustments are provided in graphical form in Figure 7.07.

FIGURE 7.07: Chart for Estimating Neutron Porosity - no shale correction

Shale and gas corrections are still needed after the lithology corrections have been applied, as described in the next section.

RECOMMENDED PARAMETERS:
None

7.09 Porosity From the Neutron Log
The response equation for the neutron porosity log also follows the classical form:

PHIN = PHIe * Sxo * PHINw (water term)
+ PHIe * (1 - Sxo) * PHINh (hydrocarbon term)
+ Vsh * PHINsh (shale term)
+ (1 - Vsh - PHIe) * Sum (Vi * PHINi) (matrix term)

WHERE:
PHINh = log reading in 100% hydrocarbon
PHINi = log reading in 100% of the ith component of matrix rock
PHIN = log reading
PHINsh = log reading in 100% shale
PHINw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)

We usually assume PHINw = PHINh = 1.0, PHINi = 0.0, and that PHINsh and Vsh are known. This results in: PHInc = PHIN - Vsh * PHINSH

If PHINi is not zero, PHIN can be adjusted as in Section 7.08; then used in the response equation. If gas is present a correction factor is sometimes applied.

After converting from old style neutron logs or adjusting for lithology, the procedure is similar to that for sonic and density, namely gas correction and shale correction.

Apply neutron shale correction.
1: PHInc = PHIN - Vsh * PHINSH

Compute neutron log gas correction.
2: IF NEUTRONGASSWITCH$ = "ON"
3: THEN PHIN = KN * PHIN

WHERE:
KD = neutron gas correction factor (fractional)
PHIN = porosity from neutron log corrected for lithology or gas (fractional)
PHInc = porosity from neutron log corrected for shale (fractional)
PHINSH = apparent neutron log porosity of 100% shale (fractional)
Vsh = volume of shale (fractional)

COMMENTS:
A chart to solve this equation, along with the lithology shifts can be found in Figure 7.08.

FIGURE 7.08: Chart for Estimating Shale Corrected Neutron Porosity

KN is in the range of 1.0 to 3.0 depending on depth of invasion, gas density and logging tool type. Use local experience. Apply this correction only if gas is known to be present and log reading is still too low after lithology corrections.

The neutron log corrected for shale is one of the least accurate methods and should only be used if no other porosity data is available. This is common for wells drilled prior to 1957 or for wells logged through casing or drill pipe.

RECOMMENDED PARAMETERS:
PHINSH is in the range 0.10 to 0.40, with a default value of 0.30.

NUMERICAL EXAMPLE:
1. Assume data from Sand "D" in Classic Example 1
PHIN = 0.28
PHINSH = 0.30
Vsh = 0.33
neutron log type = CNL
CNL / FDC units = sandstone
Rescaling is not required, as log is in correct units.
No gas correction is required.

PHInc = 0.28 - 0.33 * 0.30 = 0.18

7.10 Summary of One-Log Porosity Methods
Previous sections of this Chapter have outlined several methods for calculating porosity from the individual porosity indicating logs. They are termed one-log methods, as opposed to two-log or crossplot methods, since only a single porosity indicating log is used in each case.

These methods, may be summarized in the following generalized terms:

l. Find total porosity (PHIt) in the zone of interest by scaling the log in porosity units as indicated in Table 7.01 or by using a calculator or computer with equations detailed above.

2. Apply lithology corrections if needed.

3. Estimate apparent porosity in nearby shale (PHI_SH) by observing the log response in shales, or calculating the apparent porosity of the shale from the equations.

4. Compute shale content (Vsh) from the GR, SP or density neutron crossplots as specified in Chapter Six.

5. Derive effective porosity (PHIe) by subtracting the porosity contribution of the shale.
PHIe = PHIt - Vsh * PHI_SH

6. Apply gas corrections if needed.

WHERE:
PHIe = effective porosity (fractional)
PHI_SH = shale porosity (fractional)
PHIt = total porosity (fractional)
Vsh = shale volume (fractional)

Porosity derived from any of these methods, after all corrections are applied, is called the effective porosity.

This reduction can usually be done without the aid of a calculator and allows for an accurate visual interpretation. The technique is suitable for sonic, density or neutron logs. If results from these three methods do not agree, then the analyst must find out why. Often a poor choice of shale base lines or matrix value is at fault. Calculations should be attempted again using new parameters until a satisfactory porosity result is obtained.

Note that the gas correction suggested here is extremely inaccurate, and that these methods are not recommended in gas zones, unless sufficient outside data is available for control.

The sonic method should not be attempted if the log skips excessively, unless the log can be edited confidently. The density method must not be tried in rough or large holes, as the log cannot usually be edited accurately. Use the caliper and density correction curves as a guide. None of the methods are valid in mixed lithology, unless the lithology can be zoned by use of sample descriptions. In all cases, use appropriate matrix values for reasonable results.

The answers for Classic Example 1 from the three methods described are given in Figure 7.09. The reader should verify the results before proceeding.

FIGURE 7.09: Computed Results for Shale Corrected Porosity - Classic Example 1

Plots of the computed results from all three one-log methods for the mixed lithology example are shown in Figure 7.24. These treated the zone as a shaly sandstone, so results are poor and do not match core very well where other minerals are present.

7.11 Quick Methods for Density Neutron Crossplot Calculations
Density neutron crossplot methods involve simultaneous solution of the response equations for the two logs. The response equation for the density log in porosity units follows the classical form:

PHID = PHIe * Sxo * PHIDw (water term)
+ PHIe * (1 - Sxo) * PHIDh (hydrocarbon term)
+ Vsh * PHIDsh (shale term)
+ (1 - Vsh - PHIe) * Sum (Vi * PHIDi) (matrix term)

WHERE:
PHIDh = log reading in 100% hydrocarbon
PHIDi = log reading in 100% of the ith component of matrix rock
PHID = log reading
PHIDsh = log reading in 100% shale
PHIDw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)

The response equation for the neutron porosity log also follows the classical form:

PHIN = PHIe * Sxo * PHINw (water term)
+ PHIe * (1 - Sxo) * PHINh (hydrocarbon term)
+ Vsh * PHINsh (shale term)
+ (1 - Vsh - PHIe) * Sum (Vi * PHINi) (matrix term)

WHERE:
PHINh = log reading in 100% hydrocarbon
PHINi = log reading in 100% of the ith component of matrix rock
PHIN = log reading
PHINsh = log reading in 100% shale
PHINw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)

Various assumptions are made in order to solve these two equations simultaneously for porosity. The second variable determined by the pair is usually either of shale volume or matrix density, which can determine rock type. If one of these is chosen, the other must be assumed or previously calculated.

Very rapid methods for estimating porosity from density and neutron log data are available, based on extensive use of assumptions. They are easily memorized and should be used with mental arithmetic or calculators. The approach simulates the results from chartbooks, but more detailed formulae should be used in computers, as discussed later in this chapter.

In carbonate sequences, that is, when limestone, dolomite or anhydrite are present, place logs into limestone units. Shale correct the density and neutron logs using the effective porosity equation. Then derive the average of the two answers as shown below:
1: PHIdc = PHID - Vsh * PHIDSH
2: PHInc = PHIN - Vsh * PHINSH
3: IF PHInc < 0
4: THEN PHInc = 0

If no gas crossover exists after shale corrected, that is,
5: IF PHInc >= PHIdc
6: THEN PHIxdn = (PHInc + PHIdc) / 2

If crossover occurs when gas is affecting the logs, that is:
7: IF PHInc < PHIdc
8: THEN PHIxdn = ((PHInc ^ 2 + PHIdc ^ 2) / 2) ^ 0.5

WHERE:
PHID = density porosity log reading (fractional)
PHIdc = density log porosity corrected for shale (fractional)
PHIN = neutron porosity log reading (fractional)
PHInc = neutron log porosity corrected for shale (fractional)
PHIDSH = density log shale porosity (fractional)
PHINSH = neutron log shale porosity (fractional)
PHIxdn = porosity from density neutron crossplot (fractional)
Vsh = shale volume (fractional)

COMMENTS:
Do not gas correct density or neutron logs prior to doing this method.

Although originally published for carbonate analysis, the method is adequate for shaly sands.

If PHIdc > PHInc and gas is known NOT to be present, shale volume or PHINSH are too high.

In shaly sands, convert logs into sandstone units and enter the uncorrected density and neutron log readings into the following equations:

If no gas crossover exists, that is,
1: IF PHIN >= PHID
2: THEN PHIxdn = (PHID * PHINSH - PHIN * PHIDSH) / (PHINSH - PHIDSH)

If gas crossover is present, that is,
3: IF PHIN < PHID
4: THEN PHIxdn = ((PHIN ^ 2 + PHID ^ 2) / 2) ^ 0.5

WHERE:
PHID = density porosity log reading (fractional)
PHIdc = density log porosity corrected for shale (fractional)
PHIN = neutron porosity log reading (fractional)
PHInc = neutron log porosity corrected for shale (fractional)
PHIDSH = density log shale porosity (fractional)
PHINSH = neutron log shale porosity (fractional)
PHIxdn = porosity from density neutron crossplot (fractional)

COMMENTS:
Do not gas or shale correct the density or neutron data before using this method.

This method calculates the shale correction based on the density neutron crossplot shale volume. This method is pessimistic if heavy minerals exist in the sandstone. Since nearly every sandstone contains minerals other than quartz to some degree (such as mica, volcanic rock fragments, dolomite or calcite cement), the shaly sand model is almost ALWAYS PESSIMISTIC. An alternate method is to use the carbonate equation described earlier, placing the log data in sandstone units first. Methods listed below are also better. However, this shaly sand model has been very widely used and it can still be found in every computer-aided log analysis package. I strongly recommend NOT using this model.

An alternate quick look method for shaly sand or carbonates is:
1: PHIdc = PHID - Vsh * PHIDSH
2: PHInc = PHIN - Vsh * PHINSH
3: IF PHInc < 0
4: THEN PHInc = 0
5: PHIxdn = PHIdc + (PHInc - PHIdc) / 3

WHERE:
PHID = density porosity log reading (fractional)
PHIdc = density log porosity corrected for shale (fractional)
PHIN = neutron porosity log reading (fractional)
PHInc = neutron log porosity corrected for shale (fractional)
PHIDSH = density log shale porosity (fractional)
PHINSH = neutron log shale porosity (fractional)
PHIxdn = porosity from density neutron crossplot (fractional)
Vsh = shale volume (fractional)

COMMENTS:
This formula is effective even in gas zones and is based on the premise that the neutron log sees deeper into the zone than the density log, and thus has more gas effect.
Shale corrections in the limestone dolomite case could create apparent gas crossover and this may be real or an artifact of excessive correction. Check against known data from the well if shale correction creates crossover.

NUMERICAL EXAMPLE:
1. Data for Sand "D" in Classic Example 1 shaly sand formula:
PHID = 0.12
PHIN = 0.30
PHIDSH = 0.03
PHINSH = 0.30
PHIxdn = (0.12 * 0.30 - 0.28 * 0.03) / (0.30 - 0.03) = 0.100

The shale volume was 0.59 using this data in Chapter Six. Since this value is too high compared to the shale content from the GR, the porosity from this method will be too low.

2. If this porosity is unacceptably low, use the carbonate formula with Vsh of your choice, as shown below. Assume Vsh = 0.33 from GR, with other data as before, then:
PHInc = 0.30 - 0.33 * 0.30 = 0.20
PHIdc = 0.12 - 0.33 * 0.03 = 0.11
PHIxdn = (0.20 + 0.11) / 2 = 0.155

3. The third quick method gives, for Sand "D"
PHIxdn = 0.11 + (0.20 - 0.11) / 3 = 0.140

4. Data for Sand "C" in Classic Example 1
PHIN = 0.24
PHID = 0.37 (gas crossover)
PHIxdn = ((0.24 ^ 2 + 0.37 ^ 2) / 2) ^ 0.5 = 0.31

7.12 Shaly Sand Crossplot (Density Neutron) with Matrix Offset
This method is more long-winded than the quick method and accounts for matrix offset to an arbitrary user defined matrix value.

Reconstitute density data from density porosity log.
1: DENSm = (PHID * 1.00 + ( 1 - PHID) * (2.65 + 0.06 * (IF LOGUNIT$ = "LIMESTONE")))
* (1 + 999 * (IF DEPTHUNIT$ + "METRIC"))

Calculate density porosity for desired matrix and fluid values:
2: PHIDm = (DENSMA - DENSm) / (DENSMA - DENSW)

Calculate density offset for this matrix and fluid:
3: D = PHIDm - PHID

Calculate neutron offset for same matrix:
4: C = D - 0.25 * D * (IF NEUTRONTYPE$ = "SNP")

Calculate neutron log reading for same matrix:
5: PHINm = PHIN - C

Adjust shale values for offsets:
6: PHIDSHm = PHIDSH + D
7: PHINSHm = PHINSH - C

Calculate porosity where thers is no gas crossover, that is:
8: IF PHINm >= PHIDm
9: THEN PHIxdn = (PHIDm * PHINSHm - PHINm * PHIDSHm) / (PHINSHm - PHIDSHm)

If gas crossover occurs, or gas correction is imposed, that is:
10: IF NEUTRONTYPE$ = "SNP"
11: AND IF PHINm < PHIDm
12: OR IF GASCORRSWITCH$ = "ON"
13: THEN PHIxdn = ((PHINm ^ 2 + PHIDm ^ 2) / 2) ^ 0.5
14: IF NEUTRONTYPE$ = "CNL"
15: AND IF PHINm < PHIDm
16: OR IF GASCORRSWITCH$ = "ON"
17: THEN PHIx = - PHIDm / (PHINm / 0.8 - 1) / (1 + PHIDm / (0.8 - PHINm))
18: AND PHIxdn = PHIx + 1.8 * (0.30 - PHIx) * (DENSMA / (1 + 999 *
(IF DEPTHUNIT$ = "METRIC") - 2.65)

WHERE:
C = neutron log offset (fractional)
D = density log offset (fractional)
DENSm = density log reading (Kg/m3 or gm/cc)
DENSMA = matrix density (Kg/m3 or gm/cc)
DENSW = fluid density (Kg/m3 or gm/cc)
PHID = density log reading in zone of interest (fractional)
PHIDm = density log reading corrected for matrix offset (fractional)
PHIDSH = density log reading in 100% shale (fractional)
PHIDSHm = density log reading in 100% shale corrected for matrix offset (fractional)
PHIN = neutron log reading in zone of interest (fractional)
PHINm = neutron log reading corrected for matrix offset (fractional)
PHINSH = neutron log reading in 100% shale (fractional)
PHINSHm = neutron log reading in 100% shale corrected for matrix offset (fractional)
PHIx = density neutron porosity with matrix and gas correction applied (fractional)
PHIxdn = shaly sand crossplot porosity corrected for matrix offset

COMMENTS:
The graphical solution to these equations is shown in Figure 7.10.

FIGURE 7.10: Chart for Estimating Shale Corrected Porosity From Density Neutron Crossplot - Shaly Sand Model

Note that this method, while called the shaly sand crossplot method, will work for any lithology, providing it is constant over the computed interval (shale may vary but not the mineral mixture). Shale content is implicitly corrected by use of PHINSH and PHIDSH. The method does not use the Vsh value calculated by any other method. An alternate method is given in Section 7.13, which uses the Vsh determined independently by the analyst.

The gas correction for the SNP represents a 45 degree line on the density neutron crossplot. For the CNL, the equations represent a fan of lines originating at PHIN = 0.80 and PHID = 0.0. It is the approximate correction for a gas with specific gravity of 0.35 at the temperature and pressure usually encountered. The value of 0.80 can be varied if desired to fit a different gas density or very shallow depths.

The gas correction is usually applied automatically when crossover of the density and neutron logs occurs. In shaly or dolomitic sands, crossover may be inhibited by the nature of the lithology, so the analyst may impose the corrections if gas is known to be present. The analyst must select the matrix density towards which the gas correction should be applied. The gas correction is described more fully in Section 7.18.

The method is quite reliable as long as the density neutron separation is a function of shale content. The method can be simplified by eliminating the matrix offset calculations. They can either be ignored (at the risk of a small error in computed porosity) or by normalizing the density and neutron logs to read the same value in clean sands. How much to shift each curve to accomplish this is a trial and error procedure for the log analyst.

Because the shale volume is implied instead of an explicit input parameter, I do not recommend this method.

See Section 7.06 for additional parameters.

NUMERICAL EXAMPLE:
1. Assume data for Sand "C" in Example 1
PHID = 0.33
PHIN = 0.24 (gas crossover occurs)
DENSMA = 2680 Kg/m3
DENSW = 1000 Kg/m3
D = 0.02
C = -0.02
PHIDm = 0.33 + 0.02 = 0.35
PHINm = 0.24 - 0.02 = 0.22

If log type is SNP:
PHIxdn = ((0.22 ^ 2 + 0.35 ^ 2) / 2) ^ 0.5 = 0.30

If log type is CNL:
PHIx = - 0.35 / (0.22 / 0.8 - 1) / (1 + 0.35 / (0.8 - 0.22)) = 0.31
PHIxdn = 0.31 + (0.30 - 0.31) * (2680 / 1000 - 2.65) = 0.30

2. Assume data from Sand "D" and impose gas correction manually, since there is no gas crossover, due to shaliness.
PHID = 0.12
PHIN = 0.28
DENSMA = 2680 Kg/m3
DENSW = 1000 Kg/m3
D = 0.02
C = -0.02
PHIDm = 0.12 + 0.02 = 0.14
PHINm = 0.28 - 0.02 = 0.26
PHIDSHm = 0.03 + 0.02 = 0.05
PHINSHm = 0.30 = 0.02 = 0.28

If log type is SNP:
PHIxdn = (0.12 * 0.28 - 0.26 * 0.05) / (0.28 - 0.05) = 0.11

No gas correction was imposed. If the gas formula had been used, the result would have been:

PHIxdn = ((0.14 ^ 2 + 0.26 ^ 2) / 2 ^ 0.5 = 0.208

This is too high so the SNP should not usually be corrected in this manner.

If log type is CNL:
PHIx = - 0.14 / (0.26 / 0.8 - 1) / (1 + 0.14 / (0.8 - 0.26)) = 0.17
PHIxdn = 0.17 + 1.8 (0.30 - 0.17) * (2680 / 1000 - 2.65) = 0.18

If gas correction is not imposed and log type is CNL:
PHIxdn = (0.14 * 0.28 - 0.26 * 0.05) / (0.28 - 0.05) = 0.11

The gas correction raises the porosity by 0.07 or nearly 50% of the porosity value in this example.

7.13 Complex Lithology Crossplot (Density Neutron)
The complex lithology crossplot requires that shale corrections be applied to the density and neutron log data first. Accordingly, porosity and matrix density are derived. Log data should be put in limestone units.

Correct for shale, using the effective porosity formula on neutron and density data:
1: PHInc = PHIN - Vsh * PHINSH
2: PHIdc = PHID - Vsh * PHIDSH
3: IF PHInc < 0
4: THEN PHInc = 0

Reconstitute shale corrected density data from density porosity log:
5: DENSm = (PHIdc * 1.00 + (1 - PHIdc) * (2.65 + 0.06 * (IF LOGUNIT$ = "LIMESTONE")))
* (1 + 999 * (IF DEPTHUNIT$ = "METRIC"))

Calculate density porosity for desired matrix and fluid values:
6: PHIDm = (DENSMA - DENSm) / (DENSMA - DENSW)

Calculate density offset for this matrix and fluid:
7: D = PHIDm - PHIdc

Calculate neutron offset for same matrix:
8: C = D - 0.25 * D * (IF NEUTRONTYPE$ = "SNP")

Calculate neutron log reading for same matrix:
9: PHINm = PHInc - C

Put PHInc and PHIdc into limestone units if they are not already in this form.
10: IF LOGUNIT$ = "SANDSTONE"
11: THEN PHInc = PHInc - 0.03 - 0.01 * (IF NEUTRONTYPE$ = "SNP"
12: AND PHIdc = PHIdc + 0.03
13: LOGUNIT$ = "LIMESTONE"

Calculate pseudo matrix points and check for gas crossover based on PHINm and PHIDm. Note that PHINm and PHIDm are not used in the following porosity equation, only for the gas crossover test.

If no gas crossover, that is:
14: IF PHINm >= PHIDm
15: THEN E = 0.7 - 10 ^ ( - (1.9 + 3.1 * (IF NEUTRONTYPE$ = "CNL")) * PHInc - 0.16)
16: AND G = (2.71 - 4.00) / (2.71 - 1.00)

If gas crossover, that is,
17: IF PHINm < PHIDm
18: THEN E = - (1.17 + 2.06 PHInc + 10 ^ (-0.4 - 16.0 * PHInc))
19: AND G = 1

Calculate porosity.
If no gas crossover and tool type is SNP or CNL or gas crossover and tool type is SNP, that is:
20: IF PHINm >= PHIDm
21: AND IF NEUTRONTYPE$ = "CNL"
22: OR IF NEUTRONTYPE$ = "SNP"
23: THEN PHIxc = (G * PHInc - E * PHIdc) / (G - E)

If gas crossover, and tool type is CNL or if gas correction is imposed:
24: IF PHINm < PHIDm
25: AND IF NEUTRONTYPE$ = "CNL"
26: OR IF GASCORRSWITCH$ = "ON"
27: THEN PHIx = -PHIdc / (PHInc / 0.8 -1) / (1 + PHIdc / (0.8 - PHInc))
28: AND PHIxc = PHIx + 2 * (0.30 - PHIx) * (DENSMA / (1 + 999 * (IF DEPTHUNIT$ =
"METRIC")) - 2.71)

WHERE:
C = neutron log offset (fractional)
D = density log offset (fractional)
DENSm = density log reading after shale correction (Kg/m3 or gm/cc)
DENSMA = matrix density (Kg/m3 or gm/cc)
DENSW = fluid density (Kg/m3 or gm/cc)
E = pseudo-matrix point for neutron fractional
G = pseudo-matrix point for density (fractional)
PHID = density log reading (fractional)
PHIdc = density porosity corrected for shale (fractional)
PHIDm = density log reading corrected for matrix offset (fractional)
PHIDSH = density log shale value (fractional)
PHIN = neutron log reading (fractional)
PHINm = neutron log reading corrected for matrix offset (fractional)
PHInc = neutron porosity corrected for shale (fractional)
PHINSH = neutron log shale value (fractional)
PHIx = porosity from complex lithology crossplot corrected for shale (fractional)
PHIxc = porosity from complex lithology crossplot corrected for shale and matrix lithology (fractional)

COMMENTS:
The overall layout of the complex lithology crossplot is shown in Figure 7.11, and a detailed chart in Figure 7.12.

FIGURE 7.11: Overview Chart for Complex Lithology Model


FIGURE 7.12: Chart for Complex Lithology Porosity Model -data must be shale corrected before using the chart

Gas corrections are more fully described in Section 7.18. This is obviously a fairly complicated approach to complex lithology solution. Simplification can be achieved by eliminating the matrix shift for the gas crossover test. This is often done. With or without this simplification, complex lithology is one of the most commonly used log analysis methods for modern logs.

A streamlined version of this model can be found in Crain’s Petrophysical Pocket Pal.

See Section 7.06 for additional parameters.

NUMERICAL EXAMPLE:
1. Assume data from Sand "D"
PHID = 0.12
PHIN = 0.28
DENSMA = 2680 Kg/m3
DENSW = 1000 Kg/m3
Vsh = 0.33
PHIDSH = 0.03
PHINSH = 0.30
log units = sandstone
log type = CNL
PHInc = 0.28 - 0.33 * 0.30 = 0.18
PHIdc = 0.12 - 0.33 * 0.03 = 0.11
DENSm = (0.11 * 1.00 + (1 - 0.11) * 2.65)) * 1000 = 2468
PHIDm = (2680 - 2468) / (2680 - 1000) = 0.126
D = 0.126 - 0.11 = 0.016
C = 0.016 - 0.25 * 0.016 * 0.0 = 0.016
PHINm = 0.18 - 0.016 = 0.164
No gas crossover after matrix shift
Change log units:
PHInc = 0.18 - 0.03 - 0.01 = 0.14
PHIdc = 0.11 + 0.03 = 0.14
E = 0.7 - 10 ^ ( -(1.9 + 3.1) * 0.14 - 0.16) = 0.56
G = (2.71 - 4.00) / (2.71 - 1.00) = -0.75
PHIxc = (-0.75 * 0.14 - 0.56 * 0.14) / (-0.75 - 0.51) = 0.145

2. If Vsh was 0.59 as indicated by the density neutron method, then:
PHInc = 0.28 - 0.59 * 0.30 = 0.10
PHIdc = 0.12 - 0.59 * 0.03 = 0.10
DENSm = (0.10 * 1.00 + 1 - 0.10) * 2.65)) * 1000 = 2485
PHIDm = (2680 - 2485) / (2680 - 1000) = 0.116
D = 0.116 - 0.100 = 0.016
C = 0.016
PHINm = 0.10 - 0.016 = 0.084

PHINm is less than PHIDm, so there is gas crossover.
E = -(1.17 + 2.06 * 0.10 + 10 ^ ( -0.4 - 16.0 * 0.10)) = -1.38
G = 1.00

If log type was SNP,
PHIxc = (1.00 * 0.10 - (-1.38 * 0.10)) / (1.00 - (-1.38)) = 0.10

If log type was CNL,
PHIx = - 0.10 / (0.10 / 0.8 - 1) / (1 + 0.10 / (0.8 - 0.10)) = 0.10
PHIxc = 0.10 + 2 * (0.30 - 0.10) * (2680 / 1000 - 2.71) = 0.088

Because the user supplied matrix density is less than 2710 Kg/m3, the gas correction is negative and reduces the porosity. If a higher matrix density had been chosen, the correction could have been positive.

The shaly sand model gave a porosity of 0.11 for this shaly sand, and 0.18 when gas correction was imposed. The complex lithology model gave 0.10 and 0.088 under the same conditions. The most logical assumption to make, is that either there is no gas and the porosity is about 0.105 or that the matrix density assumption is wrong, thus causing the gas corrections to diverge from each other. In this example, raising matrix density to 2760 Kg/m3 would balance both methods.

The full blown complex lithology model shown above is a bit much for most people. This model gives results to within 0.005 porosity for most rocks.

Shale correct the density and neutron log data for each layer:
1: PHIdc = PHID - (Vsh * PHIDSH)
2: PHInc = PHIN - (Vsh * PHINSH)

PHIDSH and PHINSH are constants for each zone, and are picked only once.

Check for gas crossover after shale corrections and calculate porosity for each layer from the correct equation:
3: IF PHInc >= PHIdc, there is no gas crossover
4: THEN PHIxdn = (PHInc + PHIdc) / 2

IF gas is known to be present AND gas crossover occurs after shale corrections, apply the following gas correction:
5: IF PHInc < PHIdc, there is gas crossover
6: THEN PHIxdn = ((PHInc ^ 2 + PHIdc ^ 2) / 2) ^ 0.5

IF gas is known to be present but no crossover occurs after shale corrections, this usually means gas in dolomite or in a sandstone with lots of heavy minerals, apply the following gas correction:
7: PHIx = -PHIdc / (PHInc / 0.8 -1) / (1 + PHIdc / (0.8 - PHInc))
8: PHIxdn = PHIx + KD3 * (0.30 - PHIx) * (DENSMA / KD1 - KD2)

Where: KD1 = 1.00 for English units
KD1 = 1000 for Metric units
KD2 = 2.65 for Sandstone scale log
KD2 = 2.71 for Limestone scale log
KD3 = 1.80 for Sandstone scale log
KD3 = 2.00 for Limestone scale log

Do not use Dolomite scale log for this special case.

Computed porosity will not match core porosity unless the correct DENSMA is chosen. DENSMA should reflect the matrix density of the expected lithology. This can be predicted accurately if the PE curve can be used to determine mineral volumes in a two mineral model. Density and neutron data cannot be used for this purpose because the gas effect masks the mineral effect.

Calculate two mineral rock volumes from PE:
1: Vmin1 = (PE - PE2 - PESH * Vsh) / (PE1 - PE2)
2: Vmin2 = 1.0 - Vmin1
3: DENSma = Vmin1 * DENSMA1 + Vmin2 * DENSMA2

DENSma can be computed as a continuous curve or used as a zone parameter to replace DENSMA in equation 8.

7.14 Bulk Volume Water Crossplot (Dual Water Model)
The bulk volume water solution is an alternate density neutron method which partitions the water in the formation into the free water in the pores and the bound water in the shale. Water saturation, described in Chapter Eight, is obtained from the ratio of free water to effective pore volume. The saturation equation is still somewhat complicated.

A major assumption required is the value of the apparent density log porosity of dry clay in the shale matrix (PHIDDC). This is often a negative number because the dry clay is more dense (lower porosity) than the shale.

Calculate neutron log dry clay point from density dry clay value,
1: PHINDC = 1.00 - (1.00 - PHIDDC ) * (1.00 - PHINSH) / (1.00 - PHIDSH)

Calculate bulk volume of bound water in shale.
2: BVWSH = (PHINDC * PHIDSH - PHIDDC * PHINSH) / (PHINDC - PHIDDC)

Calculate total porosity, also called total bulk volume of water.
3: PHIt = (PHINDC * PHID - PHIDDC * PHIN) / (PHINDC - PHIDDC)

Calculate effective porosity.
4: PHIbvw = PHIt - Vsh * BVWSH

WHERE:
BVWSH = bulk volume of water attached to shale (fractional)
PHIbvw = effective porosity from BVW method (fractional)
PHID = density log reading (fractional)
PHIDDC = density dry clay point (fractional)
PHIDSH = density shale point (fractional)
PHIN = neutron log reading (fractional)
PHINDC = neutron dry clay point (fractional)
PHINSH = neutron shale point (fractional)
PHIt = total porosity (fractional)
Vsh = shale volume (fractional)

COMMENTS:
The formulae are shown graphically in Figure 7.13.

FIGURE 7.13: Chart for Dual Water Porosity Model - note that Dry Clay has a negative density porosity

This method is mathematically similar to the shaly sand crossplot method. If matrix offset is required for heavy minerals, apply the offset to all neutron and density values including shale points first, then use the above equation. The method is also called the dual water method and is the basis of many wellsite and office computer programs.

Nothing special is done in gas zones, as the values computed for PHIt and PHIe are reasonable even if gas crossover occurs.

The usual value for PHIDDC is in the range of minus 0.11 to minus 0.15. It cannot be picked by observation of logs or crossplots. It can be calculated from:
PHIDDC = (DENSDC - DENSW) / (DENSMA - DENSW).

NUMERICAL EXAMPLE:
1. Assume data for Sand "D".
PHID = 0.12
PHIN = 0.28
DENSMA = 2650 Kg/m3 (no offset)
PHIDSH = 0.03
PHINSH = 0.30

Vsh = (0.28 - 0.12 ) / (0.30 – 0.03) = 0.59
Select PHIDDC = - 0.13 by calculating dry clay porosity from dry clay density,
PHINDC = 1.00 - (1.00 - (- 0.13)) * (1.00 - 0.30) / (1.00 - 0.03) = 0.184
BVWSH = (0.184 * 0.03 - (-0.13) * 0.30) / (0.184 - (-0.13)) = 0.142
PHIt = (0.184 * 0.12 - (-0.13) * 0.28) / (0.184 - (-0.13)) = 0.186
PHIbvw = 0.186 - 0.59 * 0.14 = 0.103

7.15 Sonic Neutron Crossplot
The sonic neutron crossplot method involves the simultaneous solution of the sonic and neutron response equations for porosity. They are similar in form to the density neutron pair, and will not be repeated here (see Section 7.11 for details).

Complex lithology is best suited to this method. Since both logs respond similarly to shale, the formulae do not have much accuracy in shaly sands.

Gas effect is similar to the density neutron crossplot, so gas may be corrected for.

Calculate compaction correction for sonic.
1: CP = max (1, CDTSH / (100 - 228 * (IF DEPTHUNIT$ = "METRIC)))

Calculate sonic porosity.
2: PHIS = (DELT - DELTMA) / (DELTW - DELTMA) / CP

Calculate shale corrected porosity:
3: PHISSH = (DELTSH - DELTMA) / (DELTW - DELTMA) / CP
4: PHIsc = PHIS - Vsh * PHISSH

Calculate matrix offset (C) for neutron log, if needed, using offset formulae provided in Section 7.06. Since density data is not needed in this method, any arbitrary density porosity value can be used to calculate the matrix offset. This density porosity value should be the appropriate average porosity in the zone of interest.

Calculate neutron log value with matrix offset.
5: PHINm = PHInc - C

Check for gas crossover using PHINm and PHIsc.
If no gas crossover, that is
6: IF PHINm >= PHIsc
7: THEN E = 0.5 - 10 ^ (-5 PHInc - 0.3)
8: AND G = -0.146
9: AND PHIxsn = (G * PHInc - E * PHIsc) / (G - E)

If gas crossover occurs, that is:
10: IF PHINm < PHIsc
11: THEN PHIxsn = ((PHIsc ^ 2 + PHInc ^ 2) / 2) ^ 0.5

WHERE:
C = neutron log matrix offset (fractional)
CDTSH = sonic log shale value for compaction correction (usec/ft or usec/m)
CP = compaction factor (fractional)
DELT = sonic log reading (usec/ft or usec/m)
DELTMA = travel time in rock matrix (usec/ft or usec/m)
DELTSH = sonic log reading in shale (usec/ft or usec/m)
DELTW = travel time in water (usec/ft or usec/m)
E = neutron pseudo matrix point
G = sonic pseudo matrix point
PHIN = neutron log reading (fractional)
PHINm = neutron log value offset for matrix effect (fractional)
PHINSH = neutron log reading in shale (fractional)
PHInc = porosity from neutron log corrected for shale (fractional)
PHIS = porosity from sonic log (fractional)
PHIsc = porosity from sonic corrected for shale (fractional)
PHISSH = sonic porosity in shale (fractional)
PHIxsn = porosity from sonic neutron crossplot (fractional)
Vsh = shale volume (fractional)

COMMENTS:
The overall layout of the neutron sonic crossplot is shown in Figure 7.14 and in detail in Figure 7.15.

FIGURE 7.14: Overall Chart for Sonic Neutron Porosity Model

FIGURE 7.15: Chart for Sonic Neutron Porosity Model - shale corrected data must be entered

The gas correction represents a 45 degree line (in porosity units) on the crossplot. The method is best used in carbonates with or without gas, and is inappropriate for shaly sand.

Shear sonic data may be used in place of compressional sonic data. This is especially useful in cased holes where a shear sonic and neutron log can be run through casing. Sonic fluid and matrix parameters are in Section 7.04.

NUMERICAL EXAMPLE:
1. Assume data for Sand "D".
DELT = 300 usec/m
PHIN = 0.28
CP = 1.00
DELTSH = 328 usec/m
PHINSH = 0.30
Vsh = 0.33
DELTMA = 182 usec/m
DELTW = 616 usec/m
no matrix offset
PHIS = (300 - 182) / (616 - 182) / 1.0 = 0.27
PHISSH = (328 - 182) / (616 - 182) / 1.0 = 0.33
PHIsc = 0.27 - 0.33 * 0.33 = 0.16
PHInc = 0.28 - 0.33 * 0.30 = 0.18
E = -0.5 - 10 ^ (-5 * 0.18 - 0.3) = 0.43
G = -0.146
PHIxsn = (-0.146 * 0.18 - 0.43 * 0.16) / (-0.146 - 0.54) = 0.165

7.16 Porosity from Sonic Density Crossplot
The sonic density crossplot method involves the simultaneous solution of the sonic and density response equations for porosity. They are similar in form to the density neutron pair, and will not be repeated here (see Section 7.11 for details).

The sonic density crossplot works best in shaly sands with no gas. The resolution is poor in carbonates and gas will make the result too high. The equations are shown graphically in Figure 7.16.

Calculate compaction correction.
1: CP = max (1, CDTSH / (100 + 228 * (IF DEPTHUNIT$ = "METRIC")))

Calculate sonic shale porosity and total porosity.
2: PHISSH = (DELTSH - DELTMA) / (DELTW - DELTMA) / CP
3: PHIS = (DELT - DELTMA) / (DELTW - DELTMA) / CP

Calculate effective porosity:
4: PHIxsd = (PHID * PHISSH - PHIS * PHIDSH) / (PHISSH - PHIDSH)

WHERE:
CDTSH = sonic shale value for compaction correction (usec/ft or usec/m)
CP = compaction factor (fractional)
DELT = sonic log reading (usec/ft or usec/m)
DELTMA = travel time in rock matrix (usec/ft or usec/m)
DELTSH = sonic log reading in shale (usec/ft or usec/m)
DELTW = travel time in water (usec/ft or usec/m)
PHID = density log reading (fractional)
PHIDSH = density shale point (fractional)
PHIS = porosity from sonic log (fractional)
PHISSH = sonic porosity in shale (fractional)
PHIxsd = porosity from density sonic crossplot (fractional)

COMMENTS:
This method is pictured in Figure 7.16.

FIGURE 7.16: Chart for Sonic Density Porosity Model - shale corrected data must be entered

This method is analogous to the density neutron shaly sand method described in Section 7.06. It does not work well in mixed lithology, such as limestone, dolomite and anhydrite mixtures. This method also does not use the volume of shale (Vsh) determined by the analyst, but uses the implicit shale correction determined by the sonic shale point (PHISSH) and the density shale point (PHIDSH). The sonic density crossplot should not be used in gas zones, since both logs can read too high due to gas effect.

If a matrix offset is required for the density log, use the method described in Section 7.06.

Shear sonic data may be used in place of compressional sonic data. This is especially useful in cased holes where a shear sonic and neutron log can be run through casing. Sonic fluid and matrix parameters are in Section 7.04.

Calculate shale corrected density and sonic log readings and convert to English units.
1: PHIdc = PHID - Vsh * PHIDSH
2: DELTc = (DELT - Vsh * (DELTSH - DELTMA)) / (1 + 2.28 * (IF DEPTHUNIT$ = "METRIC"))
3: DENSc = PHIdc + (1 - PHIdc) * (2.65 + 0.06 * (IF LOGUNIT$ = "LIMESTONE"))

Calculate velocity data from sonic travel time data.
4: VELOGc = 10 ^ 6 / DELTc
5: VELMA = 10 ^ 6 / DELTMA
6: VELW = 10 ^ 6 / DELTW

Calculate sonic porosity.
7: C = 1 - (VELOGc / (VELMA * ((DENSMA / DENSc) ^ 0.5))) ^ (1 / 1.9)
8: D = DELTc ^ 2 - DENSc * (DELTMA ^ 2) / DENSMA
9: E = DENSc * (DELTW ^ 2) / DENSW - DENSc * (DELTMA ^ 2) / DENSMA
10: IF C <= 0.37
11: THEN PHIxhr = C
12: OTHERWISE PHIxhr = ((0.47 - E / D) / 0.1) * E / D + ((0.37 - C) / 0.1) * C

WHERE:
C = intermediate term
D = intermediate term
DELT = sonic log reading in zone of interest (usec/ft or usec/m)
DELTc = sonic log reading corrected for shale (usec/ft or usec/m)
DELTMA = sonic log reading in l00% matrix rock (usec/ft or usec/m)
DELTSH = sonic log reading in l00% shale (usec/ft or usec/m)
DELTW = sonic log reading in 100% water (usec/ft or usec/m)
DENSc = density log reading corrected for shale (gm/cc or Kg/m3)
E = intermediate term PHID = density log reading in zone of interest (fractional)
PHIDSH = density log reading in 100% shale (fractional)
PHIxhr = porosity from sonic log by Hunt-Raymer crossplot (fractional)
VELOGc = sonic velocity log reading corrected for shale (ft/sec or m/sec)
VELMA = sonic velocity log reading in l00% matrix rock (ft/sec or m/sec)
VELW = sonic velocity log reading in 100% water (ft/sec or m/sec)
Vsh = shale volume (fractional)

COMMENTS:
The Hunt-Raymer equations for density sonic are an extension of their work for the sonic log (see Section 7.04). The results are too high in gas and can be corrected by a proper choice of DENSW and DELTW. The method is not universally applicable and should be tested in each area before use.

RECOMMENDED PARAMETERS:
Range Default
PHIDSH -0.03 to +0.20 0.00
DELTSH (English) 75 to 140 100
DELTSH (Metric) 225 to 460 328
See Section 7.04 and 7.06 for additional parameters.

NUMERICAL EXAMPLE:
1. Sonic Density Crossplot - data from Sand "D"
DELT = 300 usec/m
PHID = 0.12
DELTSH = 328 usec/m
PHIDSH = 0.03
Vsh = 0.33
DELTW = 616
DELTMA = 182
CP = 1.00
no matrix offset
PHISSH = (328 - 182) / (616 - 182) / 1.0 = 0.33

PHIS = (300 - 182) / (616 - 182) / 1.0 = 0.27
PHIxsd = (0.12 * 0.33 - 0.27 * 0.03) / (0.33 - 0.03) = 0.105

Vsh was 0.48 using this data in Chapter Six, but Vsh from GR is only 0.33. Therefore, porosity may be too low due to the shale correction built into this method.

2. Hunt-Raymer Sonic Density Crossplot - data from Sand "D"
Vsh = 0.33
PHIdc = 0.12 - 0.33 * 0.03 = 0.11
DELTc = (300 - 0.33 * (328 - 182)) / 3.28 = 76.8 usec/ft
DENSc = 0.11 + (1 - 0.11) * 2.65 = 2.47
VELOGc = 10 ^ 6 / 76.8 = 13020 ft/sec
VELMA = 10 ^ 6 / 55.5 = 18020 ft/sec
VELW = 10 ^ 6 / 188.0 = 5320 ft/sec
C = 1 - (13020 / (18020 * ((2.65 / 2.47) ^ 0.5))) ^ (1 / 1.9) = 0.173
PHIxhr = 0.173

This is considerably higher than the standard sonic density crossplot which over corrected for shale in this example.

7.17 Summary of Crossplot Methods
Some two-log crossplot methods correct for shale content or gas when the matrix rock type is known. Others correct for shale by using one-log methods first, then correct for matrix lithology or gas. These methods cannot perform all three tasks at once. If gas is present, the rock matrix knowledge must be provided by the analyst. There are iterative two-log methods which attempt to resolve this dilemma, not covered in this Chapter.

Some restrictions that apply to one-log methods apply to two-log methods. The sonic should not be used if cycle-skipping is excessive, and the density should not be used in large or rough holes.

A crossplot method involving density log data is usually tested first to see if the method should be used at all. The test is as follows:

1: IF CAL > CALIM OR DDENS > DENSLIM

then, use one-log sonic or one-log neutron or sonic neutron crossplot method.

WHERE:
CAL = caliper log reading (inches or mm)
CALIM = caliper limited above which density log is deemed to be useless (inches or mm)
DDENS = density correction curve reading (gm/cc or kg/m3)
DENSLIM = density correction curve limit above which density log is deemed to be useless (gm/cc or kg/m3)

If the porosity answers are too high, or too many points appear to have gas crossover, then suspect a bad hole or excessive shale corrections.

The results for the crossplot methods for Example 1 are listed in Figure 7.17.

FIGURE 7.17: Computed Results for Crossplot Porosity Methods

Compare these data with that of Figure 7.09 for one-log methods. Data for the complex lithology example are presented in Figure 7.18. Note that the best match to core porosity is from the complex lithology program, which is not too surprising.

FIGURE 7.18: Computed Results for Mixed Lithology Example

7.18 Discussion of Gas Correction Methods
The gas correction used in the previously described crossplot methods vary. To summarize:

1. Gas correction is applied to density neutron shaly sand crossplot when crossover of raw density and neutron occurs on the clean matrix line defined by the analyst by his or her choice of DENSMA (matrix density).

2. Gas correction is applied to the density neutron complex lithology crossplot when crossover of shale corrected density and shale corrected neutron occurs on the clean matrix line defined by the analyst.

3. Gas correction is applied to sonic neutron complex lithology crossplot when crossover of shale corrected sonic and shale corrected neutron occurs on the clean matrix line defined by the analyst.

4. No gas correction is made for sonic density crossplot. Both log curves could be corrected for gas before entering the crossplot algorithm (by replacing DELTW with DELTGAS and DENSW with DENSGAS) but the result is likely to be inaccurate.

5. Gas correction may be applied to the above methods, if corrections are imposed and neutron log type is CNL, even if there is no crossover. This is usually required in heavy minerals such as dolomite.

THIS GAS CORRECTION SHOULD BE IMPOSED IF YOU KNOW YOU HAVE GAS, AND IF YOU KNOW THE MINERAL MIXTURE IS HEAVIER THAN LIMESTONE (DENSMA > 2710 Kg/m3), AND IF YOU KNOW THE CORRECT MATRIX DENSITY TO MAKE THE CORRECTION WITH. DO NOT USE IT IF YOU DO NOT UNDERSTAND IT.

The correction is usually needed in dolomite or dolomitic sands or high porosity shaly sands with low to moderate invasion. Gas correction should not be imposed on SNP data because it is unlikely that invasion is so shallow that correction is required.

Before the introduction of the photo-electric effect (PE) log curve, it was easy for a log analyst to miss a gas filled dolomite reservoir. The standard density neutron crossplot would show a low porosity limestone, when in fact the zone is a medium porosity dolomite. Since the density neutron looks like limestone (curve separation is small) and the PE looks like dolomite (PE near 3.0), this discrepancy is a red flag that a special case exists. Many computer programs will not trigger gas corrections unless density neutron crossover is present, and most programs do not contain explicit algorithms to handle this special case.

6. The AMOUNT of the correction is determined by the matrix density (DENSMA) in the equations. The lower the matrix density, the lower the final porosity will be and vice versa.

Examples of the results for various matrix densities are shown in Figure 7.18 with gas correction IMPOSED.

FIGURE 7.19: Computed Results for Gas Corrected Porosity from Complex Lithology Model

Since there was no crossover, and hence no automatic gas correction, the answer porosity was far too low using a standard sand matrix density of 2650 Kg/m3. The correct matrix density is between 2740 and 2780 and may vary with depth, which requires good core control and parameter zoning for best results.

The matrix density output curve will be relatively flat at the input DENSMA value if shale volume is low or shale density equals matrix density.

This correction is virtually identical to the gas correction used in Schlumberger's VOLAN program for a gas density of 0.35 gm/cc. Note that the math is not the same as VOLAN, so some difference in results should be expected.

An illustration of how the points move under different conditions is given in Figure 7.20.

FIGURE 7.20: Graph for Gas Correction on Density Neutron Crossplot Porosity

Point A goes to A1, if CNL and DENSMA = 2710 (crossover)
goes to A2, if SNP regardless of DENSMA
goes to A3, if CNL and DENSMA = 2870.

Point B goes to B1, if CNL and DENSMA = 2870 and gas correction is IMPOSED
goes to B2, if CNL and DENSMA = 27l0 and gas correction is IMPOSED
goes to B3, if CNL and DENSMA = 2650 and gas correction is IMPOSED
stops at B if gas correction is not imposed or if SNP

Point C goes to C1, if CNL and DENSMA = 2870 and gas correction is IMPOSED
goes to C2, if CNL and DENSMA = 2710 and gas correction is IMPOSED
stops at C if gas correction not imposed or if SNP

The author is indebted to Jim Hamilton of Dome Petroleum for first suggesting this approach to one pass gas corrections. The approach has proved extremely successful and has matched core on hundreds of projects in many reservoir conditions.

7.19 Porosity from Microlog
Many older wells do not have porosity logs, but may have a microlog. Porosity can be derived, but it should be calibrated against core or more modern logs.

The response equation is based on Archie's formation factor and water saturation equations, which are described more fully in Chapter Eight.

Calculate porosity from the microlog if there is positive separation.
1: IF RES2 > RES1
2: THEN PHIml = 0.614 ((RMF@FT * KML) ^ 0.61) / (R2 ^ 0.75)
3: OTHERWISE PHIml = 0

WHERE:
KML = correction factor for mud cake effect (fractional)
PHIml = porosity from microlog (fractional)
RES1 = shallow microlog (1 inch) reading (ohm-m)
RES2 = deep microlog (2in) reading (ohm-m)
RMF@FT = mud filtrate resistivity (ohm-m)

COMMENTS:
No shale correction can be applied, so use caution. Since there is seldom any positive separation in really shaly sands, these will not usually cause any problem, except understate the potential of some shaly sands.

This method works well in good hole conditions, and with medium to high porosity. It should be used only if no other porosity indicating log is available, which is common in wells drilled before 1957. More complicated programs are available which simulate the microlog butterfly chart, but this simpler formula works nearly as well.

The chart is shown in Figure 7.21 and one such program in Figure 7.22.

FIGURE 7.21: Chart for Microlog Porosity Method


FIGURE 7.22: FORTRAN Code for Microlog Porosity Method

NUMERICAL EXAMPLE:
1. Assume microlog data:
RES1 = 3 ohm-m
RES2 = 4 ohm-m
RMF@FT = 1.0 ohm-m
mud weight = 1200 Kg/m3
KML = 0.847
PHIml = 0.614 * ((1.0 * 0.847) ^ 0.61) / (4 ^ 0.75) = 0.20

7.20 Porosity From Shallow Resistivity Logs
Porosity from proximity log, microlaterolog, microspherically focused log, spherically focused log, short normal, or shallow laterolog can be determined and is often used when no other porosity log is available. It can also be used to check microlog porosity if no other check is available.

The response equation is based on Archie's formation factor and water saturation equations, which are described more fully in Chapter Eight.

1: PHIxo = (A / ((RXO / RMF) * (SXO ^ N))) ^ (l / M)

WHERE:
A = tortuosity exponent
M = cementation exponent
N = saturation exponent
PHIxo = porosity derived from shallow resistivity device (fractional)
RMF = mud filtrate resistivity (ohm-m)
RXO = resistivity from shallow resistivity device (ohm-m)
SXO = water saturation in invaded zone (fractional)

COMMENTS:
No shale corrections are applied, so use caution. This method is a last resort, since an assumption about SXO must be made. SXO cannot be calculated for this method since it requires knowledge of porosity. Shale corrected versions of this equation can be created by inverting one of the shale corrected saturation equations in Chapter Eight.

A nomograph for solving these equations is provided in Figure 7.23.

FIGURE 7.23: Chart for Shallow Resistivity Porosity Method

RECOMMENDED PARAMETERS:
Normal values for A, M, N and SXO
for sandstone A = 0.62 M = 2.15 N = 2.00
for carbonates A = 1.00 M = 2.00 N = 2.00
for water zone SXO = 1.00
for hydrocarbon zone with high porosity SXO = 0.60
for hydrocarbon zone with medium porosity SXO = 0.70
for hydrocarbon zone with low porosity SXO = 0.80
for heavy oil and tar sands, SXO = SW = 0.10 to 0.30

NUMERICAL EXAMPLE:
1. Assume shallow resistivity data:
RXO = 20 ohm-m
RMF@FT = 1.0 ohm-m
A = 0.62
M = 2.15
N = 2.00
SXO = 1.00
PHIxo = (0.62 / ((20.0 / 1.0) * (1.0 ^ 2.0))) ^ (1 / 2.15) = 0.20

2. If zone was hydrocarbon bearing, assume:
SXO = 0.70
PHIxo = (0.62 / ((20.0 / 1.0) * (0.7 ^ 2.0))) ^ (1 / 2.15) = 0.28

7.21 Porosity from Deep or Medium Resistivity Log
This method can only be applied in water bearing zones, although correction for hydrocarbon content can be made if water saturation is reasonably well known from other sources, such as offset wells or capillary pressure data.

The response equation is based on Archie's formation factor and water saturation equations, which are described more fully in Chapter Eight.

1: PHIrt = (A / ((RESD / RW@FT) * (SW ^ N))) ^ (1 / M)

WHERE:
A = tortuosity exponent
M = cementation exponent
N = saturation exponent
PHIrt = porosity from deep resistivity (fractional)
RESD = deep resistivity log reading (ohm-m)
RW@FT = formation water resistivity (ohm-m)
SW = water saturation in un-invaded zone (fractional)

COMMENTS:
No shale corrections are applied, so use caution. This method is not usually used in hydrocarbon zones and is an absolute last resort. The result is often used in a porosity playback log to look for possible hydrocarbon zones by observing the separation between PHIrt and the other porosity logs. Shale corrected methods may be created from saturation equations in Chapter Eight.

A nomograph to solve these equations is provided in Figure 7.24.

FIGURE 7.24: Chart for Deep Resistivity Porosity Method

RECOMMENDED PARAMETERS:

Normal values for A, M, N and SW:
for sandstone A = 0.62 M = 2.15 N = 2.00
for carbonates A = 1.00 M = 2.00 N = 2.00
for water zones SW = 1.00
for hydrocarbon zone with high porosity SW = 0.20
for hydrocarbon zone with medium porosity SW = 0.40
for hydrocarbons zone with low porosity SW = 0.60

NUMERICAL EXAMPLE:
1. Assume deep resistivity data:
RESD = 5.0 ohm-m
RW@FT = 0.25 ohm-m
A = 0.62
M = 2.15
N = 2.00
SW = 1.00
PHIrt = (0.62 / ((5.0 / 0.25) * (1.0 ^ 2.0)) ^ (1 / 2.15) = 0.20

If SW = 0.40
PHIrt = (0.62 / ((5.0 / 0.25) * (0.4 ^ 2.0))) ^ (1 / 2.15) = 0.46

2. This last result suggests the zone could not be hydrocarbon bearing, otherwise the RESD value was incorrectly picked. Assume RESD = 50, then;
PHIrt = (0.62 / ((50 / 0.25) * (0.4 ^ 2.0))) ^ (1 / 2.15) = 0.16

This is a more reasonable result.

7.22 Summary of Porosity From Resistivity Methods
The methods presented provide a mechanism for analyzing ancient logs by computer. Experience has shown them to work well provided some control is exercised on the mud filtrate and water resistivity values upon which they depend. This is done by comparing results to cores or more modern log suites in the same formations nearby. When presented by computer, the results will not appear graphically to be any different or any less accurate than the most sophisticated multi-log analysis. Therefore, a warning note should be annotated on the results.

These porosity methods also rely on a knowledge of SXO or SW, which cannot usually be derived accurately prior to knowing the correct porosity. Thus, if no other porosity method is available, these methods could give misleading results, with porosity being too low in hydrocarbon bearing zones.

If approximate porosity is known, water saturation (SW) can be estimated from the PHIxSW method or the resistivity ratio method described in Chapter Eight. Invaded zone saturation (SXO) can be estimated from:

SW = PHIxSW / PHIest
SXO = (SW) ^ (1 / 5)

WHERE:
PHIest = estimated effective porosity (fractional)
PHIxSW = porosity saturation product (fractional)
SW = water saturation (fractional)
SXO = invaded zone water saturation (fractional)

This approach could be more accurate than the guidelines provided earlier for estimating SW and SXO. Shale corrections are not included, so care must be exercised in shaly sands.

Normal values for porosity-saturation product are:

7.23 Non-Porous Lithology Triggers
Some sedimentary rocks have apparent porosity indicated by various logs, but are truly non-porous. Such rocks as coal, anhydrite, gypsum and salt fall into this category. In order to discriminate these rocks, and set the porosity value to zero, use specific trigger levels for each log to help identify these minerals. The number of logs required to exceed their trigger level, before the zone is described as non-porous, is also needed. It is called the lithology logic level.

The logic is as follows:

The above table is in metric units.

For example, to formulate a logical operator from the above rules for coal, select the individual trigger logic for each log type:

1: E = (RESD > RTTRIG + PHIN > NTTRIG + PHID > DNTRIG
+ DELT > DTTRIG + GR < GRTRIG)
2: IF E >= G
3: AND G # 0
4: THEN PHIe = 0
5: AND PhiFLAG$ = "C"
6: AND COALCOUNT = COALCOUNT + 1

Zone is defined as coal. Otherwise, PHIe = porosity from whichever porosity method was used in the calculation.

WHERE:
COALCOUNT= counter for number of points which were defined to be coal
DELT = sonic log reading (usec/ft or usec/m)
DNTRIG = density coal trigger (fractional)
DTTRIG = sonic coal trigger (usce/ft or usec/m)
E = sum of triggers which are exceeded
G = coal trigger level (0 = don’t check, 1 to 5 = number of triggers needed to indicate specified lithology)
GR = gamma ray log reading (API)
GRTRIG = gamma ray coal trigger (API)
NTTRIG = neutron coal trigger (fractional)
PHID = density log reading (fractional)
PHIe = effective porosity assigned to this zone
PHIN = neutron log reading (fractional)
RESD = resistivity log reading (ohm-m)
RTTRIG = resistivity coal trigger (ohm-m)

COMMENTS:
The expected lithology in a zone will determine which zero porosity rocks should be triggered. The logic level is determined by how many logs are available and how well they resolve the lithologic problem.

NUMERICAL EXAMPLE:
1. Assume the trigger levels as for coal in the above table, and data as follows:
RESD = 350 ohm-m
PHIN = 0.45
PHID = 0.45
DELT = 328 ohm-m
GR = 15
LITH = COAL
RTTRIG = 200 ohm-m
NTTRIG = 0.40
DNTRIG = 0.40
DTTRIG = 300 ohm-m
GRTRIG = 50

Then all five tests are true and zone is coal.

2. If RESD = 180, then four tests are true and one is false. Zone is coal, if G = 4 and not coal if G = 5.

3. Assume instead:
RESD = 180
PHIN = 0.30
PHID = 0.45
DELT = 300
GR = 20

Then only two tests are true and three are false. If G > 2, then the zone is not coal, and is probably a zone with bad hole conditions.

4. If G = 0; then zone is not coal, regardless of the number of true tests.

7.24 Material Balance for Porosity (Maximum Porosity)
Material balance for porosity is needed to prevent too high a porosity in very shaly sections or in bad hole conditions. It can also be used in shaly sands as a porosity method, when only a GR or SP log is available with no porosity indicating logs.

1: IF PHIe < 0
2: THEN PHIe = 0
3: AND PhiNEG = PhiNEG + 1
4: IF PHIe > (1.00 - Vsh) * PHIMAX
5: THEN PHIe = PHIMAX * (1.00 - Vsh)
6: AND PhiPOS = PhiPOS + 1

WHERE:
PHIe = porosity from any method (fractional)
Vsh = shale content from any method (fractional)
PHIMAX = maximum expected porosity in clean rock (fractional)
PhiPOS = counter for porosity greater than PHIMAX
PhiNEG = counter for porosity less than zero

COMMENTS:
Don't fail to use this routine. It makes computer output look pretty nice even with poor logs.

This material balance prevents the sum of shale volume, porosity and rock matrix from exceeding 100%.

RECOMMENDED PARAMETERS:
Normal values for PHIMAX:
Very high porosity sandstone (and tar sands) PHIMAX = 0.38
High to medium porosity PHIMAX = 0.30
Low or shaly medium porosity PHIMAX = 0.20
Very low porosity, PHIMAX = 0.10

PHIMAX is also useful for estimating porosity in shaly sands where only an SP or gamma ray log is available, since it provides a porosity value based only upon the shale content and the analyst's assumed maximum possible porosity. With offset well data or core data for control, this is not a bad approach for wells with a very limited log suite. It is often used in computer analysis of ancient logs. Because of its gross assumptions, a warning note should be annotated on the results, if the method is used.

1: PHImx = PHIMAX * (1.00 - Vsh)

WHERE:
PHImx = porosity from maximum porosity (fractional)
Vsh = shale content from any method (fractional)
PHIMAX = maximum expected porosity in clean rock (fractional)

COMMENTS:
This method has been very successful where good control of Vsh and PHIMAX are possible, based on core data and density neutron porosity in offset wells.

RECOMMENDED PARAMETERS:
Normal values for PHIMAX:
Very high porosity sandstone (and tar sands) PHIMAX = 0.38
High to medium porosity PHIMAX = 0.30
Low or shaly medium porosity PHIMAX = 0.20
Very low porosity, PHIMAX = 0.10

NUMERICAL EXAMPLE:
1. For example, if Vsh = 0.65 and PHIe = 0.30, this would mean the sand fraction has 85% porosity, which is unreasonable. The sand fraction porosity is derived as follows:
PHIsd = PHIe / (1 - Vsh) = 0.30 / (1 - 0.65) = 0.85

Using the PHIMAX equation with a PHIMAX of 0.30:
PHIe = (1 - 0.65) * 0.30 = 0.105

which indicates the sand fraction has a porosity of 0.30 by edict of the analyst. This is more reasonable than the value of 0.85 derived earlier.

7.25 Useful Porosity
There is a recent trend among petrophysicists and engineers to partition porosity into a useful and a non-useful fraction. The concept of useful porosity, as opposed to effective porosity, is helpful where very small pores exist. These tiny pores do not connect to other pores and thus do not contribute to useful reservoir volume or reservoir energy. They are invariably water filled and nothing flows from them or through them. The tiny pores are called micro porosity; the larger, more effective, pores are called macro porosity. Thus:

1: PHIuse = PHIe - PHImicro

In sandstones, micro porosity is often associated with volcanic rock fragments that are part of the sandstone mineral mixture. In carbonates, micro porosity is associated with micrite, matrix, or pin point vugs.

The quantity of micro porosity cannot always be found directly from logs but is usually assessed as a constant fraction, KM1, of the effective porosity. This constant can be found by examination of thin section visual porosity. Where micro porosity is associated with silt or a volcanic mineral (Vmin2) in a quartz sandstone:

2: KM1 = Vsilt / (Vqrtz + Vsilt)
OR 2A: KM1 = Vmin2 / (Vqrtz + Vmin2)
3: PHIuse = PHIe * (1 ? KM1)

In some cases, the micro porosity is assumed to be a constant, PHIoffset, over an interval (ie, PHImicro is not proportional to effective porosity). This appears to happen in carbonates with unconnected pinpoint vugs (PHIppv), micritic carbonates (PHImict), or carbonates with matrix porosity (PHImatr). In all three cases, PHIoffset is found by comparing visual porosity in thin sections to log analysis porosity.

4: PHIuse = PHIe - PHIoffset

In log analysis terminology, matrix porosity usually means effective porosity (PHIe). However, in petrographic (thin section) analysis, matrix porosity (PHImatr) is non-useful porosity contained in the very fine-grained matrix material deposited between the granular or crystalline rock structure.

PHIppv, PHImict, and PHImatr may be varied according to rules developed by the analyst for the zone. A crossplot of visual porosity from thin section analysis versus PHIe from logs is a useful tool for determining the appropriate correction to obtain PHIuse. Typical rules might be:

5: PHIuse = PHIe - PHIsec (This is pretty pessimistic)
6: PHIuse = PHIsec (This may be optimistic)
7: PHIuse = PHIe - KMATR * (1 - PHIe) / (1 - KMATR)
8: PHIuse = PHIe - PHIsc * KMICT / PHISavg

KMATR and KMICT would be in the range 0.01 to 0.08, averaging 0.04, and cannot exceed PHIt.

7.26 Porosity from Nuclear Magnetic Log
The Log Response Equation for modern nuclear magnetic logs is the same as for all other logs. The difference between the NMR and other porosity logs is that the Log Response Equation is solved by the service company at logging time, instead of by the analyst after the logs are delivered. This transform is illustrated in Figure 7.25.

FIGURE 7.25: Nuclear Magnetic Resonance Response to Fluids

The matrix and dry clay terms of NMR response are zero. An NMR log run today can display clay bound water (CBW), irreducible water (capillary bound water, BVI), and mobile fluids (hydrocarbon plus water, BVM), also called free fluids or free fluid index (FFI). On older logs, only free fluids (FFI) are recorded and some subtractions, based on other open hole logs, are required.

For modern NMR logs:
1: PHIt = PHIcbw + PHIbvi + PHIbvm
2: PHIe = PHIbvi + PHIbvm
3: PHIuse = PHIbvm

WHERE:
PHIcbw = clay bound water (fractional
PHIbvi = irreducible water or capillary bound water (fractional)
PHIbvm = mobile fluids (fractional)
PHIt = total porosity (fractional)
PHIe = effective porosity (fractional)
PHIuse = useful porosity (fractional)

COMMENTS
Some or all of the sums defined above may be displayed on the delivered log. Log presentation is far from standard for NMR logs.

In some situations, mobile water can be separated from hydrocarbon, and sometimes gas can be distinguished from oil, by further (experimental) processing of the original signal. However, the depth of investigation and measurement volume are tiny, so the hydrocarbon indication is from the invaded zone.

For the same reason, PHIt and PHIe from NMR do not always agree with that derived from density neutron methods, which see much larger volumes of rock.

RECOMMENDED PARAMETERS
None required.

For older NMRlogs:
1: PHIuse = FFI
2: SWir = KBUCKL / PHIuse
3: PHIe = FFI / (1 - SWir)
4: BVWSH = (PHINSH + PHIDSH) / 2
5: PHIt = PHIe + Vsh * BVWSH

WHERE:
FFI = free fluids or free fluid index (fractional)
PHIuse = useful porosity (fractional)
PHIt = total porosity (fractional)
PHIe = effective porosity (fractional)
SWir = irreducible water saturation (fractional)
Vsh = shale volume (fractional)
BVWSH = bound water in shale (fractional)

COMMENTS
PHIe and PHIt should be compared to density neutron or other methods defined earlier.

RECOMMENDED PARAMETERS
KBUCKL is in the range 0.010 to 0.100, with a default of 0.030.

7.27 Fracture Porosity
There are a number of techniques published for calculating fracture porosity from conventional open hole logs. All were developed before the processing of formation micro-scanner data for fracture aperture became common. These older methods over-estimate fracture porosity. The only correct method is to use fracture aperture and frequency data from FMI/FMS processed logs:

1: PHIfrac = 0.001 * Wf * Df * KF1

Where:
KF1 = number of main fracture directions
= 1 for sub-horizontal or sub-vertical
= 2 for orthogonal sub-vertical
= 3 for chaotic or brecciated
PHIfrac = fracture porosity (fractional)
Df = fracture frequency (fractures per meter)
Wf = fracture aperture (millimeters)

Fracture porosity is exceedingly small and seldom is larger than 0.25% (0.0025 fractional). This is well below the noise level of conventional open hole logs. Fracture aperture from cores or thin section may be exaggerated due to stress release, so be cautious using this data.

7.28 Porosity from Three-Mineral Lithology Model
This method assumes that lithology is known from a UMA - DENSMA 3 mineral model or some other method that will determine mineral volumes accurately. The method can also be used if Vmin1 and Vmin2 (and Vmin3 if desired) are derived from sonic density neutron (Mlith/Nlith or DELTMA/DENSMA), core description, or sample description (core grain density is needed for these last two).

Calculate shale density from shale porosity (a constant for each zone):
1: DENSSH = PHIDSH * KD1 + (1 - PHIDSH) * KD2

PHIDSH and DENSSH are constants for each zone, chosen from the density log in a nearby shale.

Translate density porosity for each layer to density units:
2: DENS = PHID * KD1 + (1 - PHID) * KD2

Where:
KD1 = 1.00 for English units
KD1 = 1000 for Metric units
KD2 = 2.65 for English units Sandstone scale log
KD2 = 2650 for Metric units Sandstone scale log
KD2 = 2.71 for English units Limestone scale log
KD2 = 2710 for Metric units Limestone scale log
KD2 = 2.87 for English units Dolomite scale log
KD2 = 2870 for Metric units Dolomite scale log

Calculate matrix density from lithology results:
3: DENSma = (Vmin1*DENS1 + Vmin2*DENS2 + (1 - Vmin1 - Vmin2)*DENS3)*(1 - Vsh)
+ Vsh * DENSSH

Calculate porosity from density response equation:
4: PHIped = (DENS - DENSma) / (DENSW - DENSma)

WHERE:
DENS = density log reading in zone of interest (gm/cc or Kg/m3)
DENS1 = density log reading for mineral 1 (gm/cc or Kg/m3)
DENS2 = density log reading for mineral 2 (gm/cc or Kg/m3)
DENS3 = density log reading for mineral 3 (gm/cc or Kg/m3)
DENSma = computed density log reading in 100% matrix rock (gm/cc or Kg/m3)
DENSSH = density log reading in 100% shale (gm/cc or Kg/m3)
DENSW = density log reading in 100% water (gm/cc or Kg/m3)
PHID = porosity from uncorrected density log (fractional)
PHIDSH = apparent density log porosity of 100% shale (fractional)
DENS1 = density log reading for mineral 1 (gm/cc or Kg/m3)
DENS2 = density log reading for mineral 2 (gm/cc or Kg/m3)
DENS3 = density log reading for mineral 3 (gm/cc or Kg/m3)
Vmin1 = volume of mineral 1 (fractional)\Vmin2 = volume of mineral 2 (fractional)
Vmin3 = volume of mineral 3 (fractional)
Vsh = shale volume (fractional)

COMMENTS:
Use when data is available, but use care since errors in lithology calculation are exaggerated into the porosity equation.

Do not use in bad hole conditions or in gas zones.

This method is equivalent to a 4 mineral model where one mineral is considered to be porosity. Shale, which is calculated separately, is a fifth mineral. The model can be rephrased as a two mineral model by setting V3 to zero (ie V1 + V2 = 1.0).

To calibrate to core porosity, adjust DENS1, DENS2, DENS3, DENSSH or Vsh to obtain a better match by trial and error. Appropriate crossplots may assist, or regression of PHIped vs core porosity may be used.

7.29 Selection of Porosity Method
The answers for all the porosity solutions will vary, and in some cases be unreasonable or impossible to calculate due to lack of data. In order of preference, we would choose:

1. Density neutron crossplot (if hole is good and if both logs are available).

2. Sonic density crossplot if neutron is unavailable and no gas is present.

3. Sonic neutron crossplot in carbonates or in bad hole where density is not acceptable.

4. Density log corrected for shale (in good hole only).

5. Sonic log corrected for shale (in bad hole or if nothing else is available).

6. Neutron log corrected for shale (in bad hole or if nothing else is available).

7. Microresistivity, shallow or deep resistivity as a last resort.

8. Apply maximum porosity and material balance constraints to selected method.

Discard unreasonable answers, and/or revise shale or matrix assumptions and re-compute if difference between methods is too large.

Normally you will settle upon a method that suits you and the zone under consideration. You will not have time to compute results from all methods. Use the above list as a guide to reduce your effort and to gain a better chance for success on the first pass. Log analysis is seldom satisfactory on the first pass in new areas, so do not be bashful about trying several methods.

Then keep a record of which methods worked best in which areas.

Porosity is usually reported to the nearest 1/10 of one percent (or 0.001 fractional) but this can be reduced to the nearest percent (or 0.01 fractional) for hand calculations or for porosity greater than 0.25.

7.30 Effective Porosity Routines
The simplest porosity routines are those for the one-log porosity methods.

Similar routines for DENS, PHID or PHIN can be constructed by changing the name of the first porosity algorithm and the input curve name to the appropriate value. The algorithms recommended are:

PHIdc for DENS input
PHIDm for PHID input
PHInc for PHIN input

If coal is expected to be present the COAL TRIG algorithm should be interjected between the PHIxx and PhiMAX algorithms.

The routines for more complex solutions such as crossplot methods require consideration of bad hole conditions.

The PHIxdn2 algorithm could be replaced by PHIxdn3, PHIxdn4, PHIxc, PHIbvw, or PHIxsd depending on the style of analysis desired.

For carbonates the recommended routine is as follows:

The porosity routine could be any of PHIxdn1, PHIxc, or PHIxsn. In the later case the bad hole conditions do not need to be tested since density data is not in use.

An intelligent computer program could be constructed to satisfy the rules in Section 7.10, 7.17, 7.22 and 7.26 which would generate routines such as those illustrated above.

7.31 Calibrating Porosity to Core and Sample Data
The proof of a log analysis is the degree to which the porosity matches core analysis porosity. The easiest way to check this is to plot the core analysis porosity on top of the log analysis on the same depth plot. If the overlay is quite good, as in Figure 7.26, no more needs to be done except show off the comparison and brag a bit. If the core is off depth to the log porosity, shift the core depths appropriately and re-display the results.

FIGURE 7.26: Comparison of Core Porosity with Log Analysis Porosity

If the comparison is poor, there are three choices. First make sure the core data is on depth with the logs and that each log curve is on depth with each other.

CHOICE 1 (preferred): Adjust shale, matrix, and fluid parameters in the log analysis model until a better match is achieved. This may take several attempts and may require choosing a different mathematical model or mineral assemblage.

CHOICE 2: Crossplot core porosity vs log analysis porosity, and find a regression line that corrects the log result to the core, of the form:
1: PHIcorr = K1 * PHIe + K2

WHERE:
PHIcorr = corrected porosity (fractional)
PHIe = effective porosity (fractional)
K1 = slope of regression line
K2 = intercept of regression line

The regression should be the reduced major axis (RMA) method (see Chapter Eleven) and not a simple least squares regression. RMA assumes errors occur in both axes and not just in the Y axis data. An eyeball line may be best as stray outliers can be discarded quickly. The before and after crossplots can be used to document the change. Do not use the regression unless the error is reasonably low (R-squared > 0.8 or so). CAUTION: Core data must be depth matched to logs before you do this.

CHOICE 3: Perform the regression on a single input log curve instead of on PHIe, or separately on several curves. Pick the regression with the least standard deviation or highest R-squared. This creates a new log analysis model that may be used locally instead of the universal methods described in this Chapter.

You might need a multi-variant regression to account for all the minerals and fluids, or even a Principal Components analysis to obtain a statistical solution.

There are many occasions when core analysis porosity is not available for calibration of log results. The next best data set is petrographic thin section visual porosity analysis. This usually excludes micro-porosity so a regression of thin section porosity vs log analysis porosity will give useful porosity (PHIuse) instead of PHIe. Most people like this result. Thin sections can often be made from sample chips when no core exists. Thin section samples are tiny and it is sometimes difficult to scale-up these results to the whole reservoir. A large number of samples in varying facies can give statistically meaningful results. A few samples are probably useless.

15X Magnification	100X Magnification

FIGURE 7.27: Thin Section Images

FIGURE 7.28: Typical Thin Section Point Count Analysis with primary, secondary, and non-useful porosity breakdown

Not all thin section reports are as detailed as this one. Scanning electron micrograph data (SEM) is also widely used, in the same way as thin section data.

Porosity ranges as seen by microscopic examination of samples are also used as a guide. This may be useful in the absence of more quantitative data or where rough hole conditions make log analysis ambiguous. The black bar graph in Figure 7.29 shows visual porosity spanning three ranges. Log analysis should at least see porosity in the same zones and somewhat in proportion to the variations in visual porosity values.

FIGURE: 7.29: Sample Description Log with Microscopic Visual Porosity (black bar graph in center of plot)

It is also common to calibrate one-log porosity methods to crossplot methods, again by overlay plots or regression. The calibration can then be carried to wells that do not have sufficient data for crossplot analysis.

7.32 Sensitivity Analysis
Porosity determined from a log analysis is sensitive to many factors, such as shale volume, shale parameters, matrix rock parameters, and hydrocarbon (gas) corrections. Borehole roughness and washouts also play a role. Each mathematical model gives a different answer. It is very unlikely that a perfect set of parameters could be found that would make all methods give the same results.

Since the objective of log analysis is to obtain a reasonable match to core analysis porosity, we often vary one or more of the parameters listed above until this match is achieved. Review Sections 7.10 and 7.17 to see comparisons of results from different methods with a fixed set of parameters. Section 7.18 shows a sensitivity analysis of the gas correction on density neutron porosity in heavy minerals. Section 7.31 shows how to use core and sample data to assist in calibrating log analysis porosity.

Sometimes it is worth using a spreadsheet program or a computer aided log analysis package to run a sensitivity analysis to see which combination of parameters might match core data with he least difficulty. One such analysis is shown below for a variety of rock properties. Note that PV stands for pre volume and HPV for hydrocarbon pore volume.